Math Problem Statement

If f, of, x, equals, x, cubed, plus, 7, x, squared, plus, 4, x, minus, 12f(x)=x 3 +7x 2 +4x−12, which of the following is a factor of f, of, xf(x)? Answer Multiple Choice Answers left parenthesis, x, plus, 1, right parenthesis(x+1) left parenthesis, x, minus, 1, right parenthesis(x−1) left parenthesis, x, minus, 8, right parenthesis(x−8) left parenthesis, x, minus, 2, right parenthesis(x−2)

Solution

We are tasked with finding a factor of the polynomial function f(x)=x3+7x2+4x12f(x) = x^3 + 7x^2 + 4x - 12.

To determine which of the given options is a factor of the polynomial, we can apply the Factor Theorem. The Factor Theorem states that if (xc)(x - c) is a factor of a polynomial, then f(c)=0f(c) = 0.

Step 1: Check the possible factors

The options given are:

  1. (x+1)(x + 1)
  2. (x1)(x - 1)
  3. (x8)(x - 8)
  4. (x2)(x - 2)

We will substitute the corresponding values of xx (which are 1-1, 11, 88, and 22) into the polynomial f(x)f(x) and check which one makes f(x)=0f(x) = 0.

Step 2: Evaluate f(x)f(x) at each value

  1. For x=1x = -1: f(1)=(1)3+7(1)2+4(1)12=1+7412=10f(-1) = (-1)^3 + 7(-1)^2 + 4(-1) - 12 = -1 + 7 - 4 - 12 = -10 Since f(1)0f(-1) \neq 0, (x+1)(x + 1) is not a factor.

  2. For x=1x = 1: f(1)=(1)3+7(1)2+4(1)12=1+7+412=0f(1) = (1)^3 + 7(1)^2 + 4(1) - 12 = 1 + 7 + 4 - 12 = 0 Since f(1)=0f(1) = 0, (x1)(x - 1) is a factor.

Step 3: Conclusion

Thus, the factor of f(x)f(x) is (x1)(x - 1).

Final Answer:

(x1)(x - 1)

Would you like more details or help with the steps? Here are some related questions to explore:

  1. How do you perform synthetic division to factor the polynomial further?
  2. What is the general process for using the Factor Theorem?
  3. How do you identify other possible factors of a cubic polynomial?
  4. How do you confirm whether a factorization is correct?
  5. What is the role of the Remainder Theorem in polynomial division?

Tip: Always start by checking simple values like x=1x = 1 or x=1x = -1 when trying to find factors of polynomials. These are often easy to compute and can quickly reveal a factor.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Polynomials
Factor Theorem
Cubic Functions

Formulas

f(c) = 0 to verify factors using the Factor Theorem

Theorems

Factor Theorem
Remainder Theorem

Suitable Grade Level

Grades 9-12