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Chapter 4: Problem 47

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=2 x^{3}+7 x^{2}+12 x+9 $$

Short Answer

Expert verified
The real zero is \( x = -1.5 \), and the complex zeros are \( x = -1 \pm i\sqrt{2} \).

Step by step solution

01

Recognize the Problem Type

The problem is asking us to find all zeros (or roots) of the polynomial \( P(x) = 2x^3 + 7x^2 + 12x + 9 \). This involves identifying the values of \( x \) that make \( P(x) = 0 \).
02

Use the Rational Root Theorem

According to the Rational Root Theorem, any rational root \( p/q \) of the polynomial, where the polynomial's coefficients are integers, must have \( p \) as a factor of the constant term (9) and \( q \) as a factor of the leading coefficient (2). This gives possible rational roots of \( \pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2} \).
03

Test Possible Rational Roots

We test these possible roots by substituting them into the polynomial \( P(x) \) to see if they yield 0.- Testing \( x = -1 \): \[ P(-1) = 2(-1)^3 + 7(-1)^2 + 12(-1) + 9 = -2 + 7 - 12 + 9 = 2 eq 0 \]- Testing \( x = -3 \): \[ P(-3) = 2(-3)^3 + 7(-3)^2 + 12(-3) + 9 = -54 + 63 - 36 + 9 = -18 eq 0 \]Continue testing until a root is found. Upon testing,\( x = -1.5 \) yields:\[ P(-1.5) = 2(-1.5)^3 + 7(-1.5)^2 + 12(-1.5) + 9 = 0 \]
04

Use Polynomial Division

Now that we have found one root, \( x = -1.5 \), we can perform synthetic division or polynomial long division on \( P(x) \) by \( x + 1.5 \) to find the quotient, which will be a quadratic polynomial.Performing the division will yield:\[ (2x^3 + 7x^2 + 12x + 9) \div (x + 1.5) = 2x^2 + 4x + 6 \]
05

Solve the Quadratic Equation

Now solve the quadratic equation \( 2x^2 + 4x + 6 = 0 \) using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Where \( a = 2 \), \( b = 4 \), and \( c = 6 \).\[ x = \frac{-4 \pm \sqrt{16 - 48}}{4} = \frac{-4 \pm \sqrt{-32}}{4} \]Since the discriminant is negative, the quadratic has complex roots.\[ x = \frac{-4 \pm 4i\sqrt{2}}{4} = -1 \pm i\sqrt{2} \]
06

List All Zeros of the Polynomial

The zeros of the polynomial \( P(x) = 2x^3 + 7x^2 + 12x + 9 \) are \( x = -1.5 \) and \( x = -1 \pm i\sqrt{2} \).

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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 valuable tool when dealing with polynomials, especially when you're tasked with finding rational zeros. The theorem states that if a polynomial has integer coefficients, then any possible rational root, expressed as \( \frac{p}{q} \), must have \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient. This dramatically narrows down the options for possible rational roots to test.
  • For the polynomial \( P(x) = 2x^3 + 7x^2 + 12x + 9 \), the constant term is 9 and the leading coefficient is 2.
  • This gives possible factors for \( p \) as \( \pm 1, \pm 3, \pm 9 \), and for \( q \) as \( \pm 1, \pm 2 \).
      • Insert these into \( \frac{p}{q} \) to get possible rational roots: \( \pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2} \).
Once you compile this list, you can then test each potential root by substituting it back into the polynomial equation to see which ones result in zero. This process can save a lot of time compared to arbitrarily guessing possible roots.
Complex Roots
Complex roots occur when solving a polynomial equation but end up with a negative number under the square root in the quadratic formula (known as a negative discriminant). When water explores deeper into complex numbers, remember that they are expressed in the form \( a + bi \), where \( i \) is the "imaginary unit" representing \( \sqrt{-1} \).
  • In the quadratic equation derived from dividing \( P(x) \), the expression \( 2x^2 + 4x + 6 = 0 \) results in a discriminant of -32 \( (b^2 - 4ac) \).
  • Since it's negative, this indicates complex roots.
      These roots are \( x = -1 \pm i \sqrt{2} \), where \( i \) represents the imaginary part.
Comprehending complex roots is critical because polynomials with real coefficients can sometimes have non-real roots, and it's crucial for understanding the overall structure of a polynomial's solutions.
Polynomial Division
Polynomial division is a method used to simplify complex polynomials, particularly after discovering one root, of which the polynomial is a factor. When a root is determined, you can perform polynomial long division or synthetic division to factor the polynomial further.
  • After finding the root \( x = -1.5 \), dividing the original polynomial \( P(x) \) by \( x + 1.5 \) yields a more manageable polynomial \( 2x^2 + 4x + 6 \).
  • This process simplifies the polynomial, allowing us to solve it using methods like the quadratic formula for any remaining roots.
  • The quotient polynomial is crucial because it helps reveal other zeros once the polynomial has been simplified.
This step-by-step breakdown of polynomial division is particularly helpful for understanding how to handle complex polynomials and extract simpler components.
Quadratic Formula
The quadratic formula is a centerpiece in solving quadratic equations, particularly in higher-degree polynomials once you're down to a simpler quadratic form. This formula provides a direct way to find the roots of a quadratic equation \( ax^2 + bx + c = 0 \).
  • The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • In the exercise's context, applying the quadratic formula to \( 2x^2 + 4x + 6 \) reveals complex roots.
  • Specifically, substituting \( a = 2 \), \( b = 4 \), and \( c = 6 \) results in roots \( x = -1 \pm i\sqrt{2} \).
Mastering the quadratic formula is essential since it enables you to quickly and accurately find both real and complex roots, which are crucial for fully solving and understanding polynomial functions.

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Most popular questions from this chapter

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes鈥� Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=4 x^{4}-21 x^{2}+5 $$

In Example 2 we saw that some simple rational functions can be graphed by shifting, stretching, or reflecting the graph of \(y=1 / x .\) In this exercise we consider rational functions that can be graphed by transforming the graph of \(y=1 / x^{2},\) shown on the following page. (a) Graph the function $$r(x)=\frac{1}{(x-2)^{2}}$$ by transforming the graph of \(y=1 / x^{2}\) (b) Use long division and factoring to show that the function $$s(x)=\frac{2 x^{2}+4 x+5}{x^{2}+2 x+1}$$ can be written as $$s(x)=2+\frac{3}{(x+1)^{2}}$$ Then graph \(s\) by transforming the graph of \(y=1 / x^{2}\) . (c) One of the following functions can be graphed by transforming the graph of \(y=1 / x^{2} ;\) the other cannot. Use transformations to graph the one that can be, and explain why this method doesn't work for the other one. $$p(x)=\frac{2-3 x^{2}}{x^{2}-4 x+4} \quad q(x)=\frac{12 x-3 x^{2}}{x^{2}-4 x+4}$$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes鈥� Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=2 x^{4}+15 x^{3}+31 x^{2}+20 x+4 $$

Volume of a Rocket A rocket consists of a right circular cylinder of height 20 \(\mathrm{m}\) surmounted by a cone whose height and diameter are equal and whose radius is the same as that of the cylindrical section. What should this radius be (correct to two decimal places) if the total volume is to be 500\(\pi / 3 \mathrm{m}^{3}\) ?

By the Zeros Theorem, every \(n\) th-degree polynomial equation has exactly \(n\) solutions (including possibly some that are repeated). Some of these may be real and some may be imaginary. Use a graphing device to determine how many real and imaginary solutions each equation has. (a) \(x^{4}-2 x^{3}-11 x^{2}+12 x=0\) (b) \(x^{4}-2 x^{3}-11 x^{2}+12 x-5=0\) (c) \(x^{4}-2 x^{3}-11 x^{2}+12 x+40=0\)
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