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Chapter 4: Problem 18

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{3}+4 x^{2}-3 x-18 $$

Short Answer

Expert verified
The rational zeros are 2 and -3. Factored form: \( (x - 2)(x + 3)^2 \).

Step by step solution

01

Identify Possible Rational Zeros

For polynomial \( P(x) = x^3 + 4x^2 - 3x - 18 \), use the Rational Root Theorem. The potential rational zeros are the factors of the constant term (-18) divided by the factors of the leading coefficient (1). Thus, possible zeros are \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \).
02

Test Possible Zeros

Test these possible rational zeros by evaluating \( P(x) \) at each of these values.- \( P(1) = 1^3 + 4(1)^2 - 3(1) - 18 = -16 \)- \( P(-1) = (-1)^3 + 4(-1)^2 - 3(-1) - 18 = -16 \)- \( P(2) = 2^3 + 4(2)^2 - 3(2) - 18 = 0 \)Thus, \( x = 2 \) is a zero.
03

Perform Synthetic Division

Use synthetic division to divide \( P(x) \) by \( x - 2 \), since \( 2 \) is a zero:\[\begin{array}{r|rrrr}2 & 1 & 4 & -3 & -18 \ & & 2 & 12 & 18 \\hline & 1 & 6 & 9 & 0 \\end{array}\]The remainder is 0, and the quotient is \( x^2 + 6x + 9 \).
04

Factor the Quotient

The quotient \( x^2 + 6x + 9 \) can be factored by recognizing it as a perfect square:\( x^2 + 6x + 9 = (x+3)^2 \).
05

Write the Polynomial in Factored Form

Combine the factors from the previous steps. The polynomial in factored form is:\( P(x) = (x - 2)(x + 3)^2 \).
06

List All Rational Zeros

The rational zeros of the polynomial are those values of \( x \) that make each factor equal zero. From \( (x - 2)(x + 3)^2 \), the zeros are:- \( x = 2 \) - \( x = -3 \)

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Key Concepts

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

Synthetic Division
Synthetic division is a streamlined method of dividing polynomials, especially handy when you need to divide by a linear factor like \( x - c \). It reduces the complexity of long division by focusing only on the coefficients. In our example, we identified \( x = 2 \) as a root of the polynomial using the Rational Root Theorem. With synthetic division, we used this root to divide the polynomial \( P(x) = x^3 + 4x^2 - 3x - 18 \).
  • First, we align the coefficients of the polynomial. For \( P(x) \), the coefficients are 1, 4, -3, and -18.
  • Then, we perform the synthetic division using 2 as the divisor. This involves a process of multiplication and addition to find the quotient's coefficients.
  • Finally, synthetic division gives a remainder. If this remainder is zero, \( x = 2 \) is a root, confirming our earlier finding.
The result from the division is a quadratic polynomial, \( x^2 + 6x + 9 \), which simplifies solving by factoring to completely understand the roots of the original polynomial.
Polynomial Factoring
Polynomial factoring is the process of breaking down a polynomial into simpler polynomials whose product equals the original polynomial. It's a crucial step in finding zeros of the polynomial. Once we performed synthetic division, we obtained the quadratic polynomial \( x^2 + 6x + 9 \).
  • Recognizing this quadratic as a perfect square trinomial is key. Here, the expression can be rewritten as \((x + 3)^2\).
  • Factoring allows us to express the original cubic polynomial \( P(x) \) as \((x - 2)(x + 3)^2\).
This factored form is not only simpler but also reveals more about the polynomial's behavior, especially helpful for graphing or solving inequalities.
Rational Root Theorem
The Rational Root Theorem is an essential tool for finding potential rational zeros of a polynomial. It states that any rational root, represented as \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient.
  • In the polynomial \( P(x) = x^3 + 4x^2 - 3x - 18 \), the constant term is \(-18\) and the leading coefficient is 1.
  • This gives possible rational zeros of \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \).
By testing these possibilities, we found that \( x = 2 \) is a zero. This theorem is a powerful method to shortlist and test possible zeros before more intensive techniques like synthetic division.
Perfect Square Trinomials
Perfect square trinomials are a specific kind of quadratic expression that can be expressed as the square of a binomial. These trinomials take the form \( a^2 + 2ab + b^2 = (a + b)^2 \).
In our example, after using synthetic division, the resulting quadratic factor was \( x^2 + 6x + 9 \). This can be identified as a perfect square trinomial:
  • The expression can be rewritten as \((x + 3)^2\), where \(a = x\) and \(b = 3\).
  • Recognizing this structure simplifies the factoring process significantly, allowing quicker solutions.
Recognizing and applying the formula for perfect square trinomials helps in swiftly transforming complex polynomials into more manageable forms for further analysis or solving.

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Most popular questions from this chapter

Flight of a Rocket Suppose a rocket is fired upward from the surface of the earth with an initial velocity \(v\) (measured in meters per second). Then the maximum height \(h\) (in meters) reached by the rocket is given by the function $$ H(v)=\frac{R v^{2}}{2 g R-v^{2}} $$ where \(R=6.4 \times 10^{6} \mathrm{m}\) is the radius of the earth and \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity. Use a graphing device to draw a graph of the function \(h .\) (Note that \(h\) and \(v\) must both be positive, so the viewing rectangle need not contain negative values.) What does the vertical asymptote represent physically?

Graph the rational function, and find all vertical asymptotes, x- and y-intercepts, and local extrema, correct to the nearest decimal. 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. $$ y=\frac{x^{5}}{x^{3}-1} $$

A Rational Function with No Asymptote Explain how you can tell (without graphing it) that the function $$ r(x)=\frac{x^{6}+10}{x^{4}+8 x^{2}+15} $$ has no \(x\) -intercept and no horizontal, vertical, or slant asymptote. What is its end behavior?'

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$ r(x)=\frac{x^{3}+x^{2}}{x^{2}-4} $$

Show that the polynomial does not have any rational zeros. $$ P(x)=x^{3}-x-2 $$
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