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Chapter 3: Problem 57

Find a polynomial of the specified degree that has the given zeros. Degree \(3 ; \quad\) zeros \(-1,1,3\)

Short Answer

Expert verified
The polynomial is \(x^3 - 3x^2 - x + 3\).

Step by step solution

01

Express zeros as factors

Polynomials can be expressed as a product of their factors derived from their zeros. If the polynomial has zeros \(-1\), \(1\), and \(3\), the corresponding factors are \((x + 1)\), \((x - 1)\), and \((x - 3)\).
02

Multiply the first two factors

Start by multiplying the first two linear factors: \((x + 1)(x - 1)\). This is the difference of squares formula, which results in \((x^2 - 1)\).
03

Multiply the resulting expression by the remaining factor

Now, take the result from Step 2 and multiply it by the next linear factor \((x - 3)\): \((x^2 - 1)(x - 3)\).
04

Apply distribution to multiply polynomials

Distribute the \((x - 3)\) across \((x^2 - 1)\):\[(x^2 - 1)(x - 3) = x^2(x - 3) - 1(x - 3)\].
05

Simplify the expression

Calculate the above expression step by step:- \(x^2 imes x = x^3\)- \(x^2 imes (-3) = -3x^2\)- - \(-1 imes x = -x\)- \(-1 imes (-3) = +3\).Combine these terms to get the polynomial: \(x^3 - 3x^2 - x + 3\).
06

Verify the degree and terms

Ensure all terms are combined appropriately and confirm that the polynomial order is 3, as required. The result is a cubic polynomial: \(x^3 - 3x^2 - x + 3\).

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Key Concepts

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

Zeros of a Polynomial
The zeros of a polynomial are the solutions of the polynomial equation when set equal to zero. These zeros are also referred to as roots or x-intercepts of the polynomial graph. For a polynomial function, such as the one derived from our exercise, the zeros are important because:
  • They determine where the graph of the polynomial will intersect the x-axis.
  • Each zero corresponds to a factor of the polynomial鈥攖his means that for each zero, there is a linear factor.
For example, if the given zeros are \(x = -1, x = 1, \text{and} \ x = 3\), the factors would be \(x + 1, x - 1, \ ext{and} \ x - 3\) respectively. This understanding allows us to construct a polynomial by multiplying these factors together. Identifying zeros from factors or identifying factors from zeros is a fundamental skill when working with polynomials.
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable within the polynomial expression. It tells us several important things about the polynomial:
  • The largest power of the variable indicates the number of zeros (solutions) the polynomial may have.
  • The degree of the polynomial also gives us the shape or the behavior of the polynomial graph as it extends to infinity.
In the exercise, we were tasked with finding a polynomial of degree 3, indicating a cubic polynomial. This means after building and expanding the polynomial from its linear factors, the expression must have a highest term of \(x^3\). This reflects the fact that, generally, a cubic polynomial can have up to three real zeros, corresponding to its degree.
Multiplying Polynomials
Multiplying polynomials involves expanding the product of two or more polynomials into a simplified expression. It is a crucial part of forming polynomials from their factors. Let's look at how this process works using our exercise example:
  • First, recognize each given zero is associated with a linear factor. For zeros \(-1, 1, \text{and} \ 3\), the factors are \(x + 1, x - 1, \text{and} \ x - 3\).
  • Next, multiply two factors together using distributive property or special formulas such as the difference of squares.
  • Combine the resulting product with the remaining factor(s) in a step-by-step manner.
For instance, beginning with multiplying \(x + 1\) and \(x - 1\), you use the difference of squares to get \(x^2 - 1\). Then, multiply by the last factor \(x - 3\) to achieve the polynomial of degree 3: \(x^3 - 3x^2 - x + 3\). Practicing this step-by-step multiplication ensures clarity and accuracy in forming complex polynomials.

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Most popular questions from this chapter

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{3}+4}{2 x^{2}+x-1}$$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$P(x)=8 x^{3}+10 x^{2}-39 x+9 ; \quad a=-3, b=2$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$t(x)=\frac{3 x+6}{x^{2}+2 x-8}$$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$P(x)=x^{4}-2 x^{3}-9 x^{2}+2 x+8 ; \quad a=-3, b=5$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{18}{(x-3)^{2}}$$
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