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Chapter 4: Problem 2

Find a polynomial function of degree 3 with the given numbers as zeros. $$-1,0,4$$

Short Answer

Expert verified
The polynomial is \( f(x) = x^3 - 3x^2 - 4x \).

Step by step solution

01

- Understand the Given Zeros

The given zeros are -1, 0, and 4. Since they are zeros of the polynomial, the polynomial can be written in the form where these zeros are the roots.
02

- Write the Polynomial in Factored Form

Using the given zeros, we write the polynomial as \[ f(x) = a(x + 1)(x)(x - 4) \] where \(a\) is a constant.
03

- Expand the Factored Form

First, expand \((x + 1)(x)\):\[ (x + 1)(x) = x^2 + x \]Now expand \((x^2 + x)(x - 4)\):\[ (x^2 + x)(x - 4) = x^3 - 4x^2 + x^2 - 4x = x^3 - 3x^2 - 4x \]
04

- Include the Leading Coefficient

Include the leading coefficient \(a\), the polynomial becomes: \[ f(x) = a(x^3 - 3x^2 - 4x) \] For simplicity, assume \(a = 1\).
05

- Write the Final Polynomial

With \(a = 1\), the polynomial function is: \[ f(x) = x^3 - 3x^2 - 4x \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Finding Zeros
When working with polynomial functions, finding zeros, also known as roots, is crucial. Zeros are the points where the polynomial equals zero ( f(x) = 0 ). These points can tell us a lot about the graph of the polynomial.
To find zeros, look for solutions to the equation f(x) = 0 . The solutions or roots are the x -values that make the polynomial zero.
If given the zeros directly, as in this exercise, you can immediately proceed to write the polynomial in its factored form using these values.
Factored Form
The factored form of a polynomial is a way to express it as a product of its factors. Each factor corresponds to a zero of the polynomial. For our exercise, the zeros are -1, 0, and 4.
To construct the polynomial in factored form based on these zeros, write:
f(x) = a(x + 1)(x)(x - 4)
Here, a is a constant coefficient. The term (x + 1) corresponds to the zero -1 , (x) corresponds to 0 , and (x - 4) corresponds to 4 .
The factored form helps in understanding the roots and the behavior of the polynomial more clearly before expanding it.
Expanding Polynomials
Expanding polynomials involves multiplying the factors together to write the polynomial in standard form. This makes it easier to analyze the polynomial's degree and leading coefficients.
Starting with our factored form f(x) = a(x + 1)(x)(x - 4) we first expand the terms (x + 1)(x) to get x^2 + x .
Then, expand (x^2 + x)(x - 4) :
(x^2 + x)(x - 4) = x^3 - 4x^2 + x^2 - 4x = x^3 - 3x^2 - 4x .
With the leading coefficient a included and simplified with a = 1 , the polynomial becomes:
f(x) = x^3 - 3x^2 - 4x .

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Most popular questions from this chapter

For each function: a) the maximum number of real zeros that the function can have; b) the maximum number of \(x\)-intercepts that the graph of the function can have; and c) the maximum number of turning points that the graph of the function can have. $$f(x)=-x-x^{3}$$

Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. $$f(x)=2-x^{2}$$

Use long division to find the quotient \(Q(x)\) and the remainder \(R(x)\) when \(P(x)\) is divided by \(d(x) .\) Express \(P(x)\) in the form \(d(x) \cdot Q(x)+R(x)\) $$\begin{array}{l}P(x)=2 x^{3}-3 x^{2}+x-1 \\\d(x)=x-3\end{array}$$

Find the rational zeros of the function. $$\begin{aligned} P(x)=& 2 x^{5}-33 x^{4}-84 x^{3}+2203 x^{2} \\ &-3348 x-10,080 \end{aligned}$$

Factor the polynomial function \(f(x) .\) Then solve the equation \(f(x)=0\) $$f(x)=x^{4}+11 x^{3}+41 x^{2}+61 x+30$$
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