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Chapter 3: Problem 32

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$P(x)=\frac{1}{8}\left(2 x^{4}+3 x^{3}-16 x-24\right)^{2}$$

Short Answer

Expert verified
The zeros are \(x = -\frac{3}{2}\) and \(x = 2\). No additional real zeros from \(x^2 + 2x + 4 = 0\).

Step by step solution

01

Factor the Polynomial

First, focus on factoring the inner polynomial expression \(2x^4 + 3x^3 - 16x - 24\). We will try to factor by grouping. Start by grouping the terms: \((2x^4 + 3x^3)\) and \((-16x - 24)\).
02

Factor by Grouping

From the first group \(2x^4 + 3x^3\), factor out an \(x^3\) to get \(x^3(2x + 3)\). From the second group \(-16x - 24\), factor out a \(-8\) to get \(-8(2x + 3)\). This gives us the factored expression \(x^3(2x + 3) - 8(2x + 3)\).
03

Factor Completely

Both terms share a common factor \((2x + 3)\). Factor out \((2x + 3)\) to get \((2x + 3)(x^3 - 8)\). Notice that \(x^3 - 8\) can be rewritten as a difference of cubes: \((x - 2)(x^2 + 2x + 4)\). Thus, the polynomial can be factored as \((2x + 3)(x - 2)(x^2 + 2x + 4)\).
04

Find the Zeros

To find the zeros, set each factor equal to zero: 1. \(2x + 3 = 0\) gives \(x = -\frac{3}{2}\).2. \(x - 2 = 0\) gives \(x = 2\).3. \(x^2 + 2x + 4 = 0\) is a quadratic with no real roots (the discriminant \(b^2 - 4ac = 4 - 16 = -12\) is negative). Hence, no additional real zeros.
05

Sketch the Graph

The graph of \(P(x)\) will touch the x-axis at the real zeros \(x = -\frac{3}{2}\) and \(x = 2\). Because the polynomial \(P(x)\) is squared, these intersections are points of tangency, not crossings, leading to a graph that is at least tangent to the x-axis at these points and remains non-negative, as it is a square of a polynomial.

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Key Concepts

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

Polynomial Factoring
Factoring a polynomial is similar to breaking down a math problem into smaller parts that are easier to handle. The goal is to express the polynomial as a product of simpler factors. This allows us to understand its roots more clearly.
  • Polynomials can often be factored through various methods such as grouping, using a common factor, or applying special formulas like the difference of cubes.
  • In the expression given, \[ P(x)=\frac{1}{8}(2 x^{4}+3 x^{3}-16 x-24)^{2} \]we actually need to factor the inner polynomial in order to simplify and analyze it.
  • By grouping the terms and extracting the greatest common factors, we reduced the inner polynomial to \[ (2x + 3)(x - 2)(x^2 + 2x + 4). \]
The key to successful polynomial factoring is recognizing patterns and applying algebraic identities, such as recognizing squares and cubes. These make the challenge more manageable, helping students identify solutions more easily.
Zeros of a Polynomial
The zeros, or roots, of a polynomial are the values of the variable that make the polynomial equal to zero. Finding these zeros is a crucial step in graphing and analyzing polynomial functions.
  • By setting each factor of the polynomial to zero, we can find the x-values where the polynomial touches or crosses the x-axis.
  • In the polynomial \[(2x + 3)(x - 2)(x^2 + 2x + 4),\]we set each factor equal to zero:
    • \(2x + 3 = 0\) leads to the zero \(x = -\frac{3}{2}\).
    • \(x - 2 = 0\) leads to the zero \(x = 2\).
    • The factor \(x^2 + 2x + 4 = 0\) does not provide real zeros because the discriminant \(b^2 - 4ac = -12\) is negative.
These roots tell us where the graph of the polynomial will touch the x-axis. This is particularly important for graph interpretation and solving polynomial equations.
Graph of a Polynomial
Once a polynomial is factored and its zeros are found, sketching its graph becomes more straightforward. The graph of a polynomial function shows how it behaves for all real numbers of the variable.
  • The zeros found earlier \(x = -\frac{3}{2}\) and \(x = 2\) indicate where the polynomial is zero, or touches the x-axis.
  • Since the polynomial was squared in the original problem, this alters the graph's behavior around its zeros. The squared form means roots lead to points of tangency, not crossings.
  • Therefore, as \(P(x) = \frac{1}{8}[((2x + 3)(x - 2)(x^2 + 2x + 4))]^2\), the graph does not cross the x-axis at these points but just brushes it tangentially.
Understanding the nature of these zeros and how the polynomial takes shape based on its factors, even without computing every single point, can give valuable insights into the graph's overall shape and behavior.

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Most popular questions from this chapter

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{2 x(x+2)}{(x-1)(x-4)}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{x^{2}-x-6}{x^{2}+3 x}$$

The quadratic formula can be used to solve any quadratic (or second-degree) equation. You may have wondered if similar formulas exist for cubic (third- degree), quartic (fourth-degree), and higher-degree equations. For the depressed cubic \(x^{3}+p x+q=0\) Cardano (page 296 ) found the following formula for one solution: $$x=\sqrt[3]{\frac{-q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+\sqrt[3]{\frac{-q}{2}-\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}$$ A formula for quartic equations was discovered by the Italian mathematician Ferrari in \(1540 .\) In 1824 the Norwegian mathematician Niels Henrik Abel proved that it is impossible to write a quintic formula, that is, a formula for fifth-degree equations. Finally, Galois (page 273 ) gave a criterion for determining which equations can be solved by a formula involving radicals. Use the cubic formula to find a solution for the following equations. Then solve the equations using the methods you learned in this section. Which method is easier? (a) \(x^{3}-3 x+2=0\) (b) \(x^{3}-27 x-54=0\) (c) \(x^{3}+3 x+4=0\)

Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$P(x)=x^{3}-x^{2}-x-3$$

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. $$y=\frac{2 x^{2}-5 x}{2 x+3}$$
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