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 .
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 is a factor of a polynomial, then .
Step 1: Check the possible factors
The options given are:
We will substitute the corresponding values of (which are , , , and ) into the polynomial and check which one makes .
Step 2: Evaluate at each value
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For : Since , is not a factor.
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For : Since , is a factor.
Step 3: Conclusion
Thus, the factor of is .
Final Answer:
Would you like more details or help with the steps? Here are some related questions to explore:
- How do you perform synthetic division to factor the polynomial further?
- What is the general process for using the Factor Theorem?
- How do you identify other possible factors of a cubic polynomial?
- How do you confirm whether a factorization is correct?
- What is the role of the Remainder Theorem in polynomial division?
Tip: Always start by checking simple values like or when trying to find factors of polynomials. These are often easy to compute and can quickly reveal a factor.
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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
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