Chapter 4: Q. 103 (page 195)

Construct a polynomial function f with the given characteristics:
Zeros: $- 3, 1, 4$ ; degree: 3; y-intercept: 36.

Short Answer

Expert verified

$f (x) = 3 (x + 3) (x - 1) (x - 4)$

Step by step solution

Step 1. Given information.

The following information of the polynomial is given:

Zeros: -3, 1, 4; degree: 3; y-intercept: 36

Step 2. Determining the form of the polynomial.

Since the multiplicity of each zero is 1 and the degree of the polynomial is 3, there are 3 different factors present in the polynomial. The factored form of the polynomial can be expressed as

$f (x) = a (x - z_{1}) (x - z_{2}) (x - z_{3}) - - - > (1)$ where

$a$ is the constant of proportionality and

$z_{1}, z_{2}, z_{3}$ are the zeros of the polynomial.

By substituting the zeros in equation (1), the polynomial becomes

$f (x) = a (x - (- 3)) (x - 1) (x - 4) f (x) = a (x + 3) (x - 1) (x - 4) - - - > (2)$

Step 3. Finding the complete form of the polynomial.

Since the y-intercept is 36, by putting $x = 0$ in equation (2), we get the following:

$f (0) = a (3) (- 1) (- 4) 36 = 12 a a = \frac{36}{12} a = 3$

So, the polynomial function will be

$f (x) = 3 (x + 3) (x - 1) (x - 4)$

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Construct a polynomial function f with the given characteristics:
Zeros: $- 3, 1, 4$ ; degree: 3; y-intercept: 36.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Determining the form of the polynomial.

Step 3. Finding the complete form of the polynomial.

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Construct a polynomial function f with the given characteristics:Zeros: -3,1,4; degree: 3; y-intercept: 36.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Determining the form of the polynomial.

Step 3. Finding the complete form of the polynomial.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Applied Mathematics

Calculus

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Construct a polynomial function f with the given characteristics:
Zeros: $- 3, 1, 4$ ; degree: 3; y-intercept: 36.