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Chapter 4: Problem 26

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. \(P(x)=x^{3}-64\)

Short Answer

Expert verified
The zeros are 4, -2+2i鈭�3, and -2-2i鈭�3, each with multiplicity 1.

Step by step solution

01

Recognize the Type of Polynomial

The polynomial \(P(x) = x^3 - 64\) is a difference of cubes because it can be rewritten as \(P(x) = x^3 - 4^3\). A difference of cubes has the form \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).
02

Rewrite as a Difference of Cubes

Identify \(a = x\) and \(b = 4\). Substitute into the difference of cubes formula: \(x^3 - 64 = (x - 4)(x^2 + 4x + 16)\).
03

Find the Zeros of Each Factor

Set each factor equal to zero and solve. For \(x - 4 = 0\), we get \(x = 4\).\ For the quadratic \(x^2 + 4x + 16 = 0\), use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = 4\), \(c = 16\).
04

Calculate the Discriminant and Solve

Calculate the discriminant \(b^2 - 4ac = 16 - 64 = -48\). Since the discriminant is negative, the quadratic has complex solutions. Use the quadratic formula: \[ x = \frac{-4 \pm \sqrt{-48}}{2} = \frac{-4 \pm i\sqrt{48}}{2} = \frac{-4 \pm 4i\sqrt{3}}{2} = -2 \pm 2i\sqrt{3}. \]
05

State the Zeros and Their Multiplicities

The zeros of the polynomial are \(x = 4\) with multiplicity 1, and \(x = -2 + 2i\sqrt{3}\) and \(x = -2 - 2i\sqrt{3}\) each with multiplicity 1.

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Key Concepts

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

Difference of Cubes
When you see a polynomial like \( P(x) = x^3 - 64 \), you should identify it as a **difference of cubes**. This is a special form of polynomial which can be factored using the formula:
  • \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \).
In this case, you can recognize that \( a = x \) and \( b = 4 \) because 64 is equal to \( 4^3 \). This gives us:
  • \( x^3 - 64 = (x - 4)(x^2 + 4x + 16) \).
The factor \( x - 4 \) is straightforward. It yields a real zero, \( x = 4 \). The second factor, \( x^2 + 4x + 16 \), is a quadratic that requires further analysis to find its zeros. This step reveals much about the structure of the original polynomial.
Quadratic Formula
Once you have factored out the real part, you need to examine the quadratic portion: \( x^2 + 4x + 16 \). To find the zeros, apply the **quadratic formula**:
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation. In our case, these values are:
  • \( a = 1 \), \( b = 4 \), and \( c = 16 \).
First, calculate the discriminant: \( b^2 - 4ac \). For this quadratic, it is \( 16 - 64 = -48 \), which is negative. A negative discriminant indicates that the quadratic does not have real solutions, but rather complex zeros. Utilize the quadratic formula to find the complex numbers:
  • \[ x = \frac{-4 \pm \sqrt{-48}}{2} \].
This introduces a complex number component, indicating that the zeros include an imaginary part.
Complex Zeros
Complex numbers arise naturally when dealing with negative discriminants in the quadratic formula. Here, we solve \( x^2 + 4x + 16 = 0 \) using the quadratic formula and find the discriminant \( -48 \). This is crucial because this negative value leads to an **imaginary component**:
  • \( \sqrt{-48} = 4i\sqrt{3} \), where \( i \) is the imaginary unit defined as \( \sqrt{-1} \).
Plug these into the quadratic formula:
  • \[ x = \frac{-4 \pm 4i\sqrt{3}}{2} \].
Simplifying this expression, we obtain:
  • \( x = -2 \pm 2i\sqrt{3} \).
These are the **complex zeros** of the polynomial \( P(x) = x^3 - 64 \). Complex zeros often come in conjugate pairs, as they do here because one is the "mirror" of the other across the real axis. Each complex zero has its own importance, yielding a deeper understanding of polynomial roots beyond just real numbers.

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Most popular questions from this chapter

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=-x^{3}-2 x^{2}+5 x+6 $$

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?'

A box with a square base has length plus girth of 108 in. (Girth is the distance "around" the box.) What is the length of the box if its volume is 2200 in \(^{3} ?\)

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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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