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Intermediate Algebra Functions and Graphs

Katherine Yoshiwara

Section 8.1 Polynomial Functions

Subsection 8.1.1 Introduction

As noted in the introduction to this chapter, the graphs of polynomial functions have a variety of different shapes, so they are useful for modeling curves. In addition, polynomials are easy to evaluate, so they can be used to compute approximate values for other functions, such as exponentials and logarithms. In this section we’ll learn some of the algebraic techniques for studying and using polynomials.
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Subsection 8.1.2 What are Polynomials?

We have already encountered some examples of polynomial functions. Linear functions,
f(x)=ax+b
and quadratic functions
f(x)=ax2+bx+c
are special cases of polynomial functions. Polynomials can also include terms with higher powers of the variable.
Here are some examples.
f(x)=6x3−4x2+x−2g(x)=9x5−2p(x)=x4+x2+1q(x)=2x10−x7+3x6+5x3+3x
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In general, we make the following definition.
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Definition 8.1.1. Polynomial Function.

A polynomial function has the form
f(x)=anxn+an−1xn−1+an−2xn−2+⋯+a2x2+a1x+a0
where a0, a1, a2, …, an are constants and an≠0.
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The coefficient an of the highest power term is called the lead coefficient, and the highest power is the degree of the polyomial.
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Each of the polynomials above is written in descending powers, which means that the highest-degree term comes first, and the degrees of the terms decrease from largest to smallest. Sometimes it is useful to write a polynomial in ascending powers, so that the degrees of the terms increase. For example, the polynomial f(x) above would be written as
f(x)=−2+x−4x2+6x3
in ascending powers.
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Checkpoint 8.1.2. QuickCheck 1.

What is the lead coefficient of a polynomial?
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  1. The first coefficient
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  2. The constant term
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  3. The largest coefficient
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  4. The coefficient of the highest power term
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We can evaluate polynomials just as we do for any other function.
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Example 8.1.3.

Evaluate the polynomial
P(x)=x3−0.5x2−4x+2
at x=3 and at x=−2.
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Solution.

We substitute the values given for x and follow the order of operations.
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P(3)=33−0.5(3)2−4(3)+2=12.5
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P(−2)=(−2)3−0.5(−2)2−4(−2)+2=0
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Here is a graph of the polynomial.
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x-cubed-.5xsquared-4x+2
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Subsection 8.1.3 Products of Polynomials

In Example 3.2.2 of Section 3.2 we solved a quadratic equation by factoring. As another example, because
(x−4)(x+7)=x2+3x−28
the solutions of the equation
x2+3x−28=0
are 4 and −7. But now notice that the product of the two linear polynomials x−4 and x+7 is a quadratic polynomial. Or, in other words, the product of two first-degree polynomials gives us a second-degree polynomial. In fact, whenever we multiply two or more polynomials together, we get another polynomial of higher degree.
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Example 8.1.4.

Compute the products.  [TK]
  1. (x+2)(5x3−3x2+4)
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  2. (x−3)(x+2)(x−4)
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Solution.

diagram of polynomial multiplication
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Checkpoint 8.1.5. Practice 1.

Multiply (y+2)(y2−2y+3)
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Answer.
y3−y+6
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In part (a) of the Example above, we multiplied a polynomial of degree one by a polynomial of degree three, and the product was a polynomial of degree four. In part (b) of the Example, the product of three first-degree polynomials is a third-degree polynomial.
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Degree of a Product.

The degree of a product of non-zero polynomials is the sum of the degrees of the factors. That is:
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If P(x) has degree m and Q(x) has degree n, then their product P(x)Q(x) has degree m+n.
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Example 8.1.6.

Let P(x)=5x4−2x3+6x2−x+2, and
Q(x)=3x3−4x2+5x+3.
  1. What is the degree of their product? What is the coefficient of the lead term?
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  2. Find the coefficient of the x3-term of the product.
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Solution.

  1. The degree of P is 4, and the degree of Q is 3, so the degree of their product is 4+3=7. The only degree 7 term of the product is (5x4)(3x3)=15x7, which has coefficient 15.
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  2. In the product, each term of P(x) is multiplied by each term of Q(x). How might we isolate the products that result in terms sof degree 3? We get degree 3 terms by multiplying together terms of degree 0 and 3, or of degree 1 and 2. For these polynomials, the possible combinations are:
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    P(x) Q(x) Product
    2 3x3 6x3
    −2x3 3 −6x3
    −x −4x2 4x3
    6x2 5x 30x3
    The sum of the third-degree terms of the product is
    6x3−6x3+4x3+30x3=34x3,
    with coefficient 34.
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Checkpoint 8.1.7. Practice 2.

Find the coefficient of the fourth-degree term of the product of
f(x)=2x6+2x4−x3+5x2+1
and
g(x)=x5−3x4+2x3+x2−4x−2.
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Answer.
2
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Subsection 8.1.4 Modeling with Polynomials

Polynomials are used to model many variable relationships, including volume and surface area.
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Example 8.1.9.

A closed box has a square base of length and width x inches and a height of x−6 inches, as shown at right.
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box with height x-6 inches with both length and width of x inches
  1. Write a polynomial function S(x) that gives the surface area of the box in terms of the dimensions of the base.
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  2. What is the surface area of a box of length and width 18 inches?
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  3. Write a polynomial function for the volume of the box.
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Solution.

  1. The surface area of a box is the sum of the areas of its six faces,
    S=2lh+2wh+2lw
    Substituting x for l and w, and x−6 for h gives us
    S(x)=2x(x−6)+2x(x−6)+2x2=6x2−24x
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  2. We evaluate the polynomial for x=18 to find
    S(18)=6(18)2−24(18)=1512 square inches
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  3. You probably recall that the formula for the volume of a box is
    V=lwh,
    the product of its length, width, and height. So the box has volume
    V=x⋅x(x−86)=x3−6x cubic inches.
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Checkpoint 8.1.10. Practice 3.

An empty reflecting pool is 3 feet deep. It is 8 feet longer than it is wide, as illustrated at right.
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  1. Write a polynomial function S(x) that gives the surface area of the empty pool.
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  2. Write a polynomial function V(x) for the volume of the pool.
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rectangular hole 3 ft deep, x feet wide, x+8 ft long
Answer.
  1. S(x)=x2+20x+48
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  2. V(x)=3x2+24x
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Subsection 8.1.5 Cubic Polynomials

The volume of the box in Example 9, V=x3−6x, is an example of a cubic polynomial , a polynomial of degree 3. Cubic polynomials are often used in economics to model cost functions. The cost of producing x items is an increasing function of x, but its rate of increase is usually not constant.
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Example 8.1.11.

Pegasus Printing, Ltd. is launching a new magazine. The cost of printing x thousand copies is given by
C(x)=x3−24x2+195x+250
  1. What are the fixed costs, that is, the costs incurred before any copies are printed?
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  2. Graph the cost function in the window below and describe the graph.
    Xmin=0Xmax=20Ymin=0Ymax=1500
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  3. How many copies can be printed for $1200?
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  4. What does the concavity of the graph tell you about the cost function? (See Example 5.6.3 to review concavity.)
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Solution.

  1. Fixed costs are given by C(0)=250, or $250. The fixed costs include expenses like utility bills that must be paid even if no magazines are produced.
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  2. The graph is shown in figure (a). It is increasing from a vertical intercept of 250. The graph is concave down for x<8 approximately, and concave up for x>8.
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    GC graphs of a cubic cost function
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  3. We must solve the equation
    x3−24x2+195x+250=1200
    We will solve the equation graphically, as shown in figure (b). Graph y=1200 along with the cost function, and use the intersect command to find the intersection point of the graphs, (15.319,1200). C(x)=1200 when x is about 15,319, so 15,319 copies can be printed for $1200.
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  4. Although the cost is always increasing, it increases very slowly from about x=5 to about x=11. The flattening of the graph in this interval is a result of economy of scale: By buying supplies in bulk and using time efficiently, the cost per magazine can be minimized. However, if the production level is too large, costs begin to rise rapidly again.
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Note 8.1.12.

In Example 8.1.11c, we solved a cubic equation graphically. There is a cubic formula, analogous to the quadratic formula, that allows us to solve cubic equations algebraically, but it is complicated and not often used in applications.
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Cubic polynomials are also used to model smooth curves connecting given points. Such a curve is called a cubic spline.
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Checkpoint 8.1.13. Practice 4.

Leon is flying his plane to Au Gres, Michigan. He maintains a constant altitude until he passes over a marker just outside the neighboring town of Omer, when he begins his descent for landing. During the descent, his altitude, in feet, is given by
A(x)=128x3−960x2+8000
where x is the number of miles Leon has traveled since passing over the marker in Omer.
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  1. What is Leon’s altitude when he begins his descent?
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  2. Graph A(x) in the window
    Xmin=0Xmax=5Ymin=0Ymax=8000
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  3. Use the Trace feature to discover how far from Omer Leon will travel before landing. (In other words, how far is Au Gres from Omer?)
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  4. Verify your answer to part (c) algebraically.
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Answer.
  1. 8000 ft
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  2. cubic
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  3. 5 mi
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  4. A(5)=0
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Checkpoint 8.1.14. QuickCheck 3.

Fill in the blanks to complete each statement.
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  1. We find the surface area of an object by calculating the area of each face and then them.
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  2. In economics, are often used to model cost functions.
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  3. The flattening of a cost function in its midrange is a result of .
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  4. A smooth curve connecting given points by cubic polynomials is called a .
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Subsection 8.1.6 Approximations with Polynomials

Did you ever wonder how your calculator obtains decimal values for numbers such as 1.1  or 100.8 ? One way to do it is to use a polynomial. Functions such as f(x)=x  and f(x)=10x  can be approximated by polynomials, and the higher the degree of the polynomial, the better the approximation. Because evaluating a polynomial uses only addition and multiplication, calculators and computers can perform the calculations quickly and give us decimal values with great accuracy.
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Example 8.1.15.

  1. Evaluate the polynomial
    P(x)=1+0.5x−0.125x2+0.0625x3
    at x=0.1 to find an approximation for 1.1.
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  2. Use a calculator to evaluate 1.1. How close is your approximation from part (a)?
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Solution.

  1. Substituting x=0.1 into P(x) we find
    P(0.1)=1+0.5(0.1)−0.125(0.1)2+0.0625(0.1)3=1.0488125
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  2. A calculator returns the value 1.1=1.048808848. Our approximation is accurate to four decimal places. One can use a polynomial of higher degree to obtain an approximation with greater accuracy.
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Checkpoint 8.1.16. Practice 5.

  1. Evaluate the polynomial
    P(x)=10+23x+26.51x2+20.35x3
    at x=−0.2 to find an approximation for 100.8.
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  2. Use a calculator to evaluate 100.8. How close is your approximation from part (a)?
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Answer.
  1. 6.2976
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  2. 6.30957   The approximation is accurate to two decimal places.
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Subsection 8.1.7 Special Products

As we saw in Chapter 3, many quadratic polynomials can be factored into a product of two linear expressions, and these factors give us information about the roots of a quadratic equation or the x-intercepts of its graph. The same is true of cubic polynomials, so we will take a brief look at how they can be factored.
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To begin, recall the following special products involving quadratic expressions.  [TK]
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Special Products of Binomials.

(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2(a+b)(a−b)=a2−b2
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There are also special products resulting in cubic polynomials. In the Homework problems, you will be asked to verify the following products.
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Cube of a Binomial.

  1. (x+y)3=x3+3x2y+3xy2+y3
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  2. (x−y)3=x3−3x2y+3xy2−y3
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If you become familiar with these general forms, you can use them as patterns to find specific examples of such products.
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Example 8.1.17.

Write (2w−3)3 as a polynomial.  [TK]
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Solution.

Use product 2 above, with x replaced by 2w and y replaced by 3.
(x−y)3=x3−3x2y+3xy2−y3(2w−3)3=(2w)3−3(2w)2(3)+3(2w)(3)2−33Simplify.=8w3−36w2+54w−27
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Of course, we can also expand the product simply by polynomial multiplication and arrive at the same answer.
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Checkpoint 8.1.18. Practice 6.

Write (5+x2)3 as a polynomial.
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Answer.
125+75x2+15x4+x6
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Checkpoint 8.1.19. QuickCheck 4.

Fill in the blanks.
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  1. The coefficient of the highest power term of a polynomial is called the .
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  2. The largest exponent in a polynomial is called the of the polynomial.
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  3. If the degrees of the terms decrease from largest to smallest, the polynomial is written in .
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  4. The degree of a product of two cubic polynomials is .
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Subsection 8.1.8 Factoring Cubics

Another pair of products is useful for factoring cubic polynomials. Here are two more products you will be asked to verify in the Homework problems.
(x+y)(x2−xy+y2)=x3+y3(x−y)(x2+xy+y2)=x3−y3
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Viewing these products from right to left, we have the following special factorizations for the sum and difference of two cubes.
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Factoring the Sum or Difference of Two Cubes.

  1. x3+y3=(x+y)(x2−xy+y2)
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  2. x3−y3=(x−y)(x2+xy+y2)
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Caution 8.1.20.

Note the difference between the expressions  (x+y)3  and  x3+y3 ; they are not the same! You can see this easily if you refer back to the box for the cube of a binomial. You already know that  (x+y)2  is not equivalent to  x2+y2  because  (x+y)2=x2+2xy+y2 , and in fact for any power n (except 1) the expressions  (x+y)n  and  xn+yn  are not equivalent.
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When we recognize a polynomial as a sum or difference of two perfect cubes, we can then identify the two cubed expressions and apply the formula.
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Example 8.1.21.

Factor each polynomial.
  1. 8a3+b3
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  2. 1−27h6
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Solution.

  1. This polynomial is a sum of two cubes. The cubed expressions are 2a, because (2a)3=8a3, and b. Use formula 1 as a pattern, replacing x with 2a, and y with b.
    x3+y3=(x+y)(x2−xy+y2)(2a)3+b3=(2a+b)[(2a)2−(2a)b+b2]Simplify.=(2a+b)(4a2−2ab+b2)
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  2. This polynomial is a difference of two cubes. The cubed expressions are 1, because 13=1, and 3h2, because (3h2)3=27h6. Use formula 2 above as a pattern, replacing x by 1, and y by 3h2:
    x3−y3=(x−y)(x2+xy+y2)13−(3h2)3=(1−3h2)[12+1(3h2)+(3h2)2]Simplify.=(1−3h2)(1+3h2+9h4)
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Checkpoint 8.1.22. Practice 7.

Factor 125n3−p3
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Answer.
(5n−p)(25n2+5np+p2)
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Checkpoint 8.1.23. QuickCheck 5.

Decide whether each statement is true or false.
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  1. We cannot factor the sum of two squares.
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  2. We cannot factor the sum of two cubes.
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  3. (x+y)3=x3+y3
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  4. To factor a3−b3, we must first identify a and b.
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Subsection 8.1.9 An Application of Cubic Splines

Investigation 8.1.1. Bézier Curves.

A Bézier curve is actually a sequence of short curves pieced together. Each piece has two endpoints and (for nonlinear curves) at least one control point. The control points do not lie on the curve itself, but they determine its shape. Two polynomials define the curve, one for the x-coordinate and one for the y-coordinate.
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A. Linear Bézier Curves
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The linear Bézier curve for two endpoints, (x1,y1) and (x2,y2), is the straight line segment joining those two points. The curve is defined by the two functions
x=f(t)=x1⋅(1−t)+x2⋅ty=g(t)=y1⋅(1−t)+y2⋅t
for 0≤t≤1.
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  1. Find the functions f and g defining the linear Bézier curve joining the two points (−4,7) and (20). Simplify the formulas defining each function.
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  2. Fill in the table of values and plot the curve.
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    t 0 0.25 0.5 0.75 1
    x 000 000 000 000 000
    y
    8 by 8 grid
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B. Quadratic Bézier Curves: Drawing a Simple Numeral 7
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The quadratic Bézier curve is defined by two endpoints, (x1,y1) and (x3,y3), and a control point, (x2,y2).
x=f(t)=x1⋅(1−t)2+2x2⋅t(1−t)+x3⋅t2y=g(t)=y1⋅(1−t)2+2y2⋅t(1−t)+y3⋅t2
for 0≤t≤1.
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  1. Find the functions f and g for the quadratic Bézier curve defined by the endpoints (−4,7) and (2,0), and the control point (0,5). Simplify the formulas defining each function.
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  2. Fill in the table of values and plot the curve.
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    t 0 0.25 0.5 0.75 1
    x 000 000 000 000 000
    y
    8 by 8 grid
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  3. Draw a line segment from (−4,7) to (4,7) on the grid above to complete the numeral 7.
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  4. We can adjust the curvature of the diagonal stroke of the 7 by moving the control point. Find the functions f and g for the quadratic Bézier curve defined by the endpoints (4,7) and (0,−7), and the control point (0,−3). Simplify the formulas defining each function.
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  5. Fill in the table of values and plot the curve.
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    t 0 0.25 0.5 0.75 1
    x 000 000 000 000 000
    y
    8 by 8 grid
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  6. Draw a line segment from (−4,7) to (4,7) on the grid above to complete the numeral 7.
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  7. On your graphs in steps (5) and (8), plot the three points that defined the curved section of the numeral 7, then connect them in order with line segments. How does the position of the control point change the curve?
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C. Cubic Bézier Curves: Drawing a Letter y
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A cubic Bézier curve is defined by two endpoints, (x1,y1) and (x4,y4), and two control points, (x2,y2) and (x3,y3).
x=f(t)=x1⋅(1−t)2+3x2⋅t(1−t)2+3x3⋅t2(1−t)+x4⋅t3y=g(t)=y1⋅(1−t)2+3y2⋅t(1−t)2+3y3⋅t2(1−t)+y4⋅t3
for 0≤t≤1.
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  1. Find the functions f and g for the cubic Bézier curve defined by the endpoints (4,7) and (−4,−5), and the control points (3,3) and (0,−8). Simplify the formulas defining each function.
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  2. Fill in the table of values and plot the curve.
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    t 0 0.25 0.5 0.75 1
    x 000 000 000 000 000
    y
    8 by 8 grid
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  3. Connect the four given points in order using three line segments. How does the position of the control points affect the curve? Finish the letter y by including the linear Bézier curve you drew for step (2).
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Exercises 8.1.10 Problem Set 8.1

Warm Up

1.
Complete the formulas for the special products of quadratic polynomials.
  1. (a+b)2=
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  2. (a−b)2=
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3.
Use the formulas to factor the polynomials, if possible.
  1. x2−14x+49
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  2. x2+64
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  3. x2+6x+9
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  4. x2−64
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4.
Use the formulas to write down the products without multiplying:
  1. (5x−6y)2
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  2. (2x3+y5)2
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  3. (3x−8y2)(3x+8y2)
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  4. (x2+y2)2
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Skills Practice

5.
Which of the following are polynomials? If it is not a polynomial, explain why not.
  1. 5x4−3x2
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  2. 3x+1+2x2
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  3. 12a2+5a−6
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  4. 23t2+14t3+58
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7.
Write each polynomial in descending powers of x.
  1. x−1.9x3+6.4
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  2. 6xy−2x2+2y3
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8.
Add or subtract the polynomials.
  1. (2y3−4y2−y)+(6y2+2y+1)
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  2. (5x3+3x2−4x+8)−(2x3−4x−3)
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9.
Without performing the multiplication, give the degree of each product.
  1. (x2−4)(3x2−6x+2)
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  2. (x−3)(2x−5)(x3−x+2)
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  3. (3x2+2x)(x3+1)(−2x2+8)
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Exercise Group.
For Problems 10 and 11, multiply.
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Exercise Group.
For Problems 12 and 13, use the formulas for the cube of a binomial to expand the products.
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Exercise Group.
For Problems 14 and 15, write each product as a polynomial and simplify.
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Exercise Group.
For Problems 16 – 21, factor completely.
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22.
Evaluate each polynomial for n=10. Try to do the calculations mentally. What do you notice?
  1. 5n2+6n+7
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  2. 5n3+n2+3n+3
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  3. n3+1
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  4. 8n4+8n
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Applications

23.
A large wooden box (without lid) is 3 feet longer than it is wide, and its height is 2 feet shorter than its width.
  1. If the width of the box is w, write expressions for its length and its height.
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  2. Write a polynomial for the volume of the box.
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  3. Write a polynomial for the surface area of the box.
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24.
A paper company plans to make boxes without tops from sheets of cardboard 12 inches wide and 16 inches long. The company will cut out four squares of side x inches from the corners of the sheet and fold up the edges as shown in the figure.
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box
  1. Write expressions in terms of x for the length, width, and height of the resulting box.
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  2. Write a formula for the volume, V, of the box as a function of x.
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  3. What are the largest and smallest reasonable values for x?
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  4. Make a table of values for V(x) on its domain.
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  5. Graph your function V in the window
    Xmin=0Xmax=6Ymin=0Ymax=200
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  6. Use your graph to find the value of x that will yield a box with maximum possible volume. What is the maximum possible volume?
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25.
  1. A grain silo is built in the shape of a cylinder with a hemisphere on top (see the figure). Write an expression for the volume of the silo in terms of the radius and height of the cylindrical portion of the silo.
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  2. If the total height of the silo is five times its radius, write a polynomial function V(r) in one variable for its volume.
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grain silo
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26.
The annual profit, P(t), of the Enviro Company, in thousands of dollars, is given by
P(t)=2t3−152t2+3400t+30
where t is the number of years since 1980, the first year that the company showed a profit.
  1. Graph P(t) in the window
    Xmin=0Xmax=94Ymin=0Ymax=50,000
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  2. What was the profit in 1980? In 2000? In 2020?
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  3. How did the profit change from 1980 to 1981? From 2000 to 2001? From 2020 to 2021?
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  4. During which years did the profit decrease from one year to the next?
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27.
A doctor who is treating a heart patient wants to prescribe medication to lower the patient’s blood pressure. The body’s reaction to this medication is a function of the dose administered. If the patient takes x milliliters of the medication, his blood pressure should decrease by R=f(x) points, where
f(x)=3x2−13x3
  1. For what values of x is R=0?
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  2. What values of x make sense for this function? Why?
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  3. Graph the function in the window
    Xmin=0Xmax=10Ymin=0Ymax=40
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  4. How much should the patient’s blood pressure drop if he takes 2 milliliters of medication?
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  5. What is the maximum drop in blood pressure that can be achieved with this medication?
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  6. There may be risks associated with a large change in blood pressure. How many milliliters of the medication should be administered to produce half the maximum possible drop in blood pressure?
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28.
The water level (in feet) at a harbor is approximated by the polynomial
W(t)=0.00733t4−0.332t2+9.1
where t is the number of hours since the high tide. The approximation is valid for −4≤t≤4. (A negative value of t corresponds to a number of hours before the high tide.)
  1. Graph the polynomial for −4≤t≤4.
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  2. What is the water level at high tide?
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  3. What is the water level 3 hours before high tide?
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  4. When is the water level below 8 feet?
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  5. When is the water level above 7 feet?
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29.
During an earthquake, Nordhoff Street split in two, and one section shifted up several centimeters. Engineers created a ramp from the lower section to the upper section. In the coordinate system shown in the figure below, the ramp is part of the graph of
y=f(x)=−0.00004x3−0.006x2+20
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ramp
  1. By how much did the upper section of the street shift during the earthquake?
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  2. What is the horizontal distance from the bottom of the ramp to the raised part of the street?
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30.
Expand each expression by removing parentheses. What do you notice?
  1. x[x(x+3)+4]+1
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  2. x(x[x(x−7)−5]+8)−3
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  3. Evaluate each expression for x=2. Can you do this mentally? Is it easier to evaluate the expression before or after expanding it?
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  4. Use a calculator to evaluate each expression for x=0.8.
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31.
Find the first three terms of the product in ascending powers. (Do not compute the entire product!)
  1. (2−x+3x2)(3+2x−x2+2x4)
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  2. (1−2x2−x4)(4+x2−2x4)
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32.
Find the indicated term in each product. (Do not compute the entire product!)
  1. (4+2x−x2)(2−3x+2x2); x2
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  2. (3x+x3−7x5)(1+4x−3x2); x3
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33.
Verify the formula for the cube of a sum:
(x+y)3=x3+3x2y+3xy2+y3
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34.
Verify the formula for the cube of a difference:
(x−y)3=x3−3x2y+3xy2−y3
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35.
Verify the formula for factoring the sum of two cubes:
(x+y)(x2−xy+y2)=x3+y3
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36.
Verify the formula for factoring the difference of two cubes:
(x−y)(x2+xy+y2)=x3−y3
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