Applications of Zeros

Remainder Theorem

When dividing polynomials, you divide one polynomial (the dividend—the number being divided) by another (the divisor—does the dividing), and get a quotient and a remainder. The remainder can be zero, a constant, or a polynomial whose degree is less than the degree of the divisor. You can always check your answer by multiplying using the steps below:

(QUOTIENT)X(DIVISOR)+REMAINDER=DIVIDEND
Problem: 45/4 = 11 R1
so 45 divided by 4 is 11 remainder 1
(11x4) + 1 = 45
11: quotient
4 divisor
1 remainder
45 dividend
The Remainder Theorem:
If a polynomial function, f(x), is divided by some factor x -c, then the remainder is f(c).

The Remainder Theorem shown here says that the remainder will be the same answer as plugging the number into the polynomial for x and solving. In other words, if we divide a polynomial by x-c, the remainder will be the same as f(c). We extend this to the factor theorem. If the remainder after plugging in a number is zero, that means that number is a zero of the polynomial. In turn, it can be written as a factor:

Example: Determine if (x - 3) is a factor of the polynomial below.

equation image indicator

If (x-3) is a factor of the polynomial, that means 3 is the zero (set the factor equal to 0). Now, use synthetic division with 3.

3|. 1. -2 -9 18
| 3 3 -18
------------------
1 1 -6 0

Since the remainder came out to be 0, we can say that 3 is a zero of the polynomial and (x - 3) is a factor.

Determine if the following are factors of the polynomials.

The following videos will give you more examples of how to use zeros: 12 Videos to Watch on this Topic

Rough Sketch Graphs

If we have the zeros of a function, we can create a rough sketch of the graph. The zeros are the x-intercepts, so we can graph those numbers on the x-axis.

Also, the constant in a polynomial equation is the y-intercept. (The y-intercept is the value where x = 0.)

Polynomial graphs have basic shapes, depending on the degree, as shown in the table.

Polynomial Function	Degree	Graph
Constant	0	Horizontal line
Linear	1	Line with slope
Quadratic	2	Parabola
Cubic	3	"S" shape curve
Quartic	4	"W" shaped

Since the graph of a polynomial function is a continuous smooth curve, by plotting the intercepts, we can draw a rough sketch of the graph.

Multiplicity

Now, let's explore "zeros" in more detail. What happens when a "zero" occurs more than one time? This is called "multiplicity." Simply put, multiplicity is the number of times a zero occurs. If equation image indicator $LaTeX: \left(x-r\right)^m$ is a factor of P(x), then r is a zero of multiplicity m of the function. Furthermore, if m is an odd number, then the graph crosses the x-axis at (r, 0) and if m is an even number then the graph is tangent to the x-axis or touches the x-axis at (r, 0).

Look at the graph of the polynomial function equation image indicator $LaTeX: P\left(x\right)=5\left(x-2\right)^2\left(x-3\right)$ below.

The root 2 has a multiplicity of 2 and the root 3 has a multiplicity of 1. Also notice that since the factor (x-2) has an even exponent the graph is tangent to the x-axis at 2. Similarly, since the factor (x-3) has an odd exponent, the graph crosses the x-axis at 3. In other words, the power of a factor determines if the graph goes through the x axis or bounces off the axis. Even numbered powers bounce off and odd numbered powers go through.

image of a curved line that hits the x-axis twice; arrows are pointing to the x-axis intersections

We will learn more about graphing in a later lesson.

Application of Polynomials

Example 1

image of empty cardboard box A manufacturer needs a box that will hold 720 cubic inches of material.

For transporting, the box needs to be 5 inches longer than it is wide and 20 inches high. What are the dimensions of the box?

Solution:

What we can gather from problem:

width = x
Length = x+5 (5 longer)
Height = 20
Volume = 720

Formula: V = L•W• H

Solving:

equation image indicator $LaTeX: 720=\left(x\right)\left(x+5\right)\cdot20 \\ 720=20x^2+100x \\ 0=20x^2+100x+720 \\ 0=20(x^2+5x-36) \\ 0=20(x+9)(x-4) \\ x=-9, 4$

Since x is a width, it can't be -9.

So the dimensions are W = 4 in, L = 9 in and H = 20 in.

Example 2

image of man holding target above head George shoots an arrow in the air. The height of the arrow is given by the function $LaTeX: h\left(t\right)=-3t^2+18t+6$ where h is the height (in feet) and t is the time (in seconds). At what times would the arrow be 21 feet high in the air?

Put 21 in for the h(t):

equation image indicator LaTeX: 21=-3t^2+18t+6

Solve:

equation image indicator $LaTeX: 21=3t^2+18t+6 \\ 0=-3t^2+18t-15 \\ 0=-3(t^2-6t+5) \\ 0=-3(t-1)(t-5) \\ t-1=0\qquad t-5=0 \\ t=1 \qquad t=5$

Therefore, the arrow would be 21 ft high in 1 sec. and in 5 sec. (Think about the path of the arrow and why there are two times that would work.)

Example 3

You've probably used a polynomial in your head more than once when shopping. For example, you might want to know how much three pounds of flour, two dozen eggs and three quarts of milk cost. Before you check the prices, construct a simple polynomial, letting "f" denote the price of flour, "e" denote the price of a dozen eggs and "m" the price of a quart of milk. It looks like this: equation image indicator LaTeX: 3f+2e+3m . This basic algebraic expression is now ready for you to input prices. If flour costs $3.49, eggs cost $4.59 per dozen and milk costs $2.89 per gallon, you will be charged $LaTeX: 3\left(3.49\right)+2\left(4.59\right)+3\left(2.89\right)=10.47+9.18+8.67=$28.32$ at checkout, plus tax.

Example 4

The compound interest expression equation image indicator $LaTeX: P\left(1+r\right)^n$ has three variables: P, r, and n, each representing a different physical quantity. As written, this expression has only one term, consisting of two factors, P and (1 + r)ⁿ. The first factor depends only on P, while the second depends on r and n.

The following videos show more examples on Solving Polynomial Application Problems.

Application of Zeros Self-Assessment

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