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Chapter 3: Problem 17

All the real zeros of the given polynomial are integers. Find the zeros, and write the polynomial in factored form. $$P(x)=x^{3}+3 x^{2}-4$$

Short Answer

Expert verified
The zeros are 1 and -2, and the factored form is \((x-1)(x+2)^2\).

Step by step solution

01

Understanding the Problem

We are given the polynomial \( P(x) = x^3 + 3x^2 - 4 \) and need to find its real integer zeros and express it in factored form.
02

Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational solution of \( P(x) = 0 \) is a factor of the constant term, \(-4\). Thus, possible integer zeros are \( \pm 1, \pm 2, \pm 4 \).
03

Test Possible Zeros

Evaluate \( P(x) \) for each possible zero: - \( P(1) = 1^3 + 3 \times 1^2 - 4 = 0 \), so \( x = 1 \) is a zero.- \( P(-1) = (-1)^3 + 3 \times (-1)^2 - 4 = -2 \)- \( P(2) = 2^3 + 3 \times 2^2 - 4 = 16 \)- \( P(-2) = (-2)^3 + 3 \times (-2)^2 - 4 = 0 \), so \( x = -2 \) is a zero.- \( P(4) = 4^3 + 3 \times 4^2 - 4 = 84 \)- \( P(-4) = (-4)^3 + 3 \times (-4)^2 - 4 = 36 \)The integers \( 1 \) and \( -2 \) are zeros of the polynomial.
04

Division of Polynomial

Divide \( P(x) \) by \( (x-1) \) and \( (x+2) \) to find the remaining factor.First, divide \( x^3 + 3x^2 - 4 \) by \( (x-1) \):\( x^3 + 3x^2 - 4 = (x-1)(x^2 + 4x + 4) \)Now check if \( x^2 + 4x + 4 \) can be factored further.
05

Factor Quadratic

The quadratic \( x^2 + 4x + 4 \) is factorable into \( (x+2)^2 \) as it can be rewritten using the identity \((a+b)^2 = a^2 + 2ab + b^2\).So \( x^2 + 4x + 4 = (x+2)(x+2) \).
06

Write Polynomial in Factored Form

The completely factored form of the polynomial \( P(x) = x^3 + 3x^2 - 4 \) is \( (x-1)(x+2)^2 \).

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Key Concepts

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

Rational Root Theorem
The Rational Root Theorem is a handy tool when we want to find possible rational solutions to polynomial equations. It states that any rational root of the polynomial equation \( P(x) = 0 \) must be a fraction \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient. In our case, the constant term is \(-4\) and the leading coefficient is \(1\), meaning possible rational roots are just the integer factors of \(-4\): \( \pm 1, \pm 2, \pm 4 \). Using this theorem significantly narrows down the testing needed, saving us a lot of guesswork when finding zeros of a polynomial.
Polynomial Division
Polynomial division, similar to long division with numbers, helps in breaking down polynomials into simpler components. It's particularly useful when we have identified a root, \( x = r \), and want to divide the polynomial by \( (x - r) \).
To perform polynomial division, we divide the original polynomial by the poly factor corresponding to the identified root. For example, to divide \( x^3 + 3x^2 - 4 \) by \( x-1 \), we would:
  • Determine the leading term that multiplies with \( x \) to match the highest degree of the polynomial.
  • Multiply and subtract from the polynomial.
  • Repeat these steps for each subsequent term.
Hence, after division, the polynomial was transformed into \( (x-1)(x^2 + 4x + 4) \), ready for further factorization.
Factored Form
Factored form presents a polynomial as a product of its factors and offers insights into its roots. Once we鈥檝e found all roots of a polynomial, writing the polynomial in its factored form makes it easier to solve for \( x \).
In our example, we used the known factors \( (x-1) \) and \( (x+2) \). After factoring the remainder \( x^2 + 4x + 4 \) into \( (x+2)^2 \), the complete factored form of the original polynomial \( P(x) \) became \( (x-1)(x+2)^2 \).
This form highlights all integer zeros, including multiplicities, and allows for simple evaluations and understanding of the behavior of the polynomial function across different domains.

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Most popular questions from this chapter

At a certain vineyard it is found that each grape vine produces about 10 lb of grapes in a season when about 700 vines are planted per acre. For each additional vine that is planted, the production of each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by $$A(n)=(700+n)(10-0.01 n)$$ where \(n\) is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest tenth. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{4}-3 x^{3}+x^{2}-3 x+3}{x^{2}-3 x}$$g

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$s(x)=\frac{x^{2}-2 x+1}{x^{3}-3 x^{2}}$$

Graphing Quadratic Functions A quadratic function \(f\) is given. (a) Express \(f\) in standard form. (b) Find the vertex and \(x\) and \(y\) -intercepts of \(f .\) (c) Sketch a graph of \(f .\) (d) Find the domain and range of \(f\). $$f(x)=-4 x^{2}-12 x+1$$

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest tenth. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{4}}{x^{2}-2}$$
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