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Chapter 3: Problem 92

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $$f(x)=x^{3}-x^{2}-8 x+12$$

Short Answer

Expert verified
The zeros are \( x = 2 \) (multiplicity 2) and \( x = -3 \).

Step by step solution

01

- Understand the Polynomial

The polynomial given is a cubic function: \[ f(x) = x^3 - x^2 - 8x + 12 \]. The goal is to find all complex zeros of this polynomial.
02

- Use the Rational Root Theorem

The Rational Root Theorem helps identify possible rational zeros by taking the factors of the constant term (12) and dividing them by the factors of the leading coefficient (1). Possible rational zeros are: 卤1, 卤2, 卤3, 卤4, 卤6, 卤12.
03

- Test Possible Rational Zeros

Plug in the possible rational zeros into the polynomial to see if any result in zero. \[ f(1) = 1^3 - 1^2 - 8(1) + 12 = 1 - 1 - 8 + 12 = 4 \] (not a zero), \[ f(-1) = (-1)^3 - (-1)^2 - 8(-1) + 12 = -1 - 1 + 8 + 12 = 18 \] (not a zero), ... Test until finding a zero.
04

- Find a Valid Zero

When testing \( f(2) \), get \[ f(2) = 2^3 - 2^2 - 8(2) + 12 = 8 - 4 - 16 + 12 = 0 \]. Thus, 2 is a valid zero.
05

- Perform Polynomial Division

Divide the polynomial \( x^3 - x^2 - 8x + 12 \) by \( x - 2 \) (since 2 is a zero). Using synthetic division, reduce the polynomial: \[ (x^3 - x^2 - 8x + 12) \rightarrow (x - 2)(x^2 + ax + b) \].
06

- Simplify the Polynomial

Perform synthetic division to get: \[ x^3 - x^2 - 8x + 12 = (x - 2)(x^2 + x - 6) \].
07

- Factor the Quadratic Polynomial

Factor the quadratic term: \[ x^2 + x - 6 \] into \[ (x + 3)(x - 2) \]. Now, the polynomial is: \[ f(x) = (x - 2)(x + 3)(x - 2) \].
08

- List All Zeros

The zeros of the polynomial function are: \( x = 2 \) (with multiplicity 2) and \( x = -3 \).

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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 useful tool for finding potential rational zeros of a polynomial. It states that any rational solution, or zero, of a polynomial equation with integer coefficients will be a fraction \( \frac{p}{q} \), where
  • \( p \) is a factor of the constant term (the term without a variable)
  • \( q \) is a factor of the leading coefficient (the coefficient of the term with the highest power)
For the polynomial \( f(x) = x^3 - x^2 - 8x + 12 \), the constant term is 12, and the leading coefficient is 1. So, the possible rational zeros are obtained by taking the factors of 12 and dividing them by the factors of 1.

These possible rational zeros are 卤1, 卤2, 卤3, 卤4, 卤6, and 卤12. To find which of these, if any, are actual zeros, we will test each one by substituting these values into the polynomial.
synthetic division
Synthetic division simplifies the process of dividing a polynomial by a binomial of the form \( x - c \). It uses less notation than long division and is especially useful when verifying potential zeros.

To perform synthetic division:
  • Write down the coefficients of the polynomial.
  • Place the zero you are testing (in our case, 2 for the zero we found) to the left of the coefficients.
  • Bring down the leading coefficient.
  • Multiply this leading coefficient by the test zero and place the product beneath the next coefficient.
  • Add the columns vertically and repeat the multiplication and addition process until all coefficients have been used.
For the polynomial \( x^3 - x^2 - 8x + 12 \), using synthetic division with the zero 2:

  • Write the coefficients: 1, -1, -8, 12
  • Perform the operations for synthetic division with 2
  • Resulting polynomial: \( x^2 + x - 6 \)

    • You now have reduced \( x^3 - x^2 - 8x + 12 \) to \( (x - 2)(x^2 + x - 6) \).
factoring polynomials
Factoring polynomials involves expressing a polynomial as the product of its simpler polynomial factors. This step comes after using synthetic division to reduce the polynomial.

We are focusing on factoring the quadratic polynomial found through synthetic division: \( x^2 + x - 6 \). To factor this, we need to find two binomials whose product gives the original quadratic. Here鈥檚 the factorization process:
  • The factors must multiply to the constant term (-6).
  • The factors must add to the coefficient of the linear term (1).
The factors of -6 that add up to 1 are 3 and -2. Therefore, the polynomial \( x^2 + x - 6 \) factors to \( (x + 3)(x - 2) \). Then, we rewrite the original polynomial as \( (x - 2)(x + 3)(x - 2) \).
multiplicity of zeros
Multiplicity refers to the number of times a particular zero appears in the polynomial. If a zero has a multiplicity greater than 1, it means that the factor associated with this zero is repeated that many times in the factorization.

For the polynomial \( f(x) = (x - 2)(x + 3)(x - 2) \):
  • The zero 2 has a multiplicity of 2 because the factor \( (x - 2) \) appears twice.
  • The zero -3 has a multiplicity of 1 because the factor \( (x + 3) \) appears once.
Understanding the multiplicity of zeros helps in sketching the graph of the polynomial, as a zero with a higher multiplicity affects the graph鈥檚 shape more significantly at the root. Hence, the zeros of this polynomial are: \( x = 2 \) with multiplicity 2, and \( x = -3 \) with multiplicity 1.

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