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Chapter 8: Problem 127

Find a polynomial function of degree 3 whose real zeros are \(-5,-2,\) and \(2 .\) Use 1 for the leading coefficient.

Short Answer

Expert verified
The polynomial is \rP(x) = x^3 + 5x^2 - 4x - 20.

Step by step solution

01

Identify the Zeros

The given zeros of the polynomial are \r\(-5\), \(-2\), and \(2\).
02

Form the Factors

For each zero \(x = a\), a factor of the polynomial is \(x - a\). Therefore, the factors corresponding to the zeros \(-5\), \(-2\), and \(2\) are \r\((x + 5)\), \((x + 2)\), and \((x - 2)\).
03

Construct the Polynomial

The polynomial function can be constructed by multiplying the factors together: \r\(P(x) = (x + 5)(x + 2)(x - 2)\).
04

Expand the Polynomial

Expand \((x + 5)(x + 2)(x - 2)\) step-by-step.\rFirst, multiply \((x + 2)\) and \((x - 2)\):\r\((x + 2)(x - 2) = x^2 - 4\).\rThen multiply the result by \(x + 5\):\r\((x + 5)(x^2 - 4)\).
05

Distribute and Simplify

Distribute \(x + 5\) across \(x^2 - 4\):\r\(x(x^2 - 4) + 5(x^2 - 4) = x^3 - 4x + 5x^2 - 20\).\rCombine like terms:\r\(P(x) = x^3 + 5x^2 - 4x - 20\).

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Key Concepts

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

degree of polynomial
The **degree of a polynomial** is the highest power of the variable in the polynomial expression. For instance, a polynomial given by a function like \( P(x) = x^3 + 5x^2 - 4x - 20 \) has a degree of 3. This degree tells us several important things:
  • The shape of the polynomial function's graph.
  • The maximum number of real zeros the polynomial can have.
  • The general behavior or trend of the polynomial function as \( x \) approaches infinity or negative infinity.
In the given problem, you identified a polynomial of degree 3 because you're told the polynomial is supposed to have three zeros.
real zeros
**Real zeros** of a polynomial are the values of \( x \) where the polynomial equals zero. In other words, these are the points where the graph intersects the x-axis. Suppose a polynomial has three real zeros; these points would be the solutions to \( P(x) = 0 \). For the polynomial function \( P(x) = x^3 + 5x^2 - 4x - 20 \), the zeros are given as \( -5, -2, \text{ and } 2 \). Each zero corresponds to a factor of the polynomial such that setting each factor to zero gives the zeros. For example:
  • For \( x = -5 \), the factor is \( (x + 5) \).
  • For \( x = -2 \), the factor is \( (x + 2) \).
  • For \( x = 2 \), the factor is \( (x - 2) \).
constructing polynomials
When you **construct a polynomial** given its zeros, you create factors of the form \( (x - a) \) for each zero \( a \). For the polynomial with real zeros \( -5, -2, \text{ and } 2 \), the factors become \( (x + 5) \), \( (x + 2) \), and \( (x - 2) \).

The next step involves multiplying these factors together to construct your polynomial:
\[ P(x) = (x + 5)(x + 2)(x - 2) \]
Expanding these will give you the polynomial in standard form. First, multiply \((x + 2)\) and \((x - 2)\) to get \( x^2 - 4 \). Then, multiply the result by \( x + 5 \) to obtain:
\[ P(x) = (x + 5)(x^2 - 4) = x^3 + 5x^2 - 4x - 20 \]
Thus, you have successfully constructed the polynomial.
factor theorem
The **Factor Theorem** is a crucial concept stating that \( x - a \) is a factor of a polynomial \( P(x) \) if and only if \( P(a) = 0 \). This theorem helps identify factors and zeros of a polynomial quickly. Using our example, we can verify the Factor Theorem:
  • For the zero \( -5 \), substitute \( x = -5 \) in \( P(x) \):
    \[ P(-5) = (-5)^3 + 5(-5)^2 - 4(-5) - 20 = -125 + 125 + 20 - 20 = 0 \]
    Hence, \( x + 5 \) is a factor.
  • Repeat for other zeros (\(-2\) and \( 2 \)): \( x + 2 \text{ and } x - 2 \) are also factors.
The Factor Theorem not only verifies that we have the correct factors but also confirms that these factors correspond to the polynomial's real zeros.

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Most popular questions from this chapter

One, Two, Three (a) Show that \(\tan \left(\tan ^{-1} 1+\tan ^{-1} 2+\tan ^{-1} 3\right)=0\). (b) Conclude from part (a) that $$ \tan ^{-1} 1+\tan ^{-1} 2+\tan ^{-1} 3=\pi $$

Establish each identity. $$ \frac{\cos (\alpha+\beta)}{\cos (\alpha-\beta)}=\frac{1-\tan \alpha \tan \beta}{1+\tan \alpha \tan \beta} $$

Area under a Curve The area under the graph of \(y=\frac{1}{1+x^{2}}\) and above the \(x\) -axis between \(x=a\) and \(x=b\) is given by $$ \tan ^{-1} b-\tan ^{-1} a $$ (a) Find the exact area under the graph of \(y=\frac{1}{1+x^{2}}\) and above the \(x\) -axis between \(x=0\) and \(x=\sqrt{3}\). (b) Find the exact area under the graph of \(y=\frac{1}{1+x^{2}}\) and above the \(x\) -axis between \(x=-\frac{\sqrt{3}}{3}\) and \(x=1\)

Find the exact value of each expression. $$ \tan \left(\sin ^{-1} \frac{4}{5}+\cos ^{-1} 1\right) $$

Find the value of \(a\) so that the line \(a x-3 y=10\) has slope 2 .
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