Math Problem Statement
Find the value of K so the binomial (x-3) is a factor of the function f(x) = x^3 + Kx^2 + x + 6.
Solution
To determine the value of such that the binomial is a factor of the cubic polynomial , we use the Factor Theorem.
The Factor Theorem states: If is a factor of , then .
Here, because the factor is . Substituting into , we get:
Step 1: Simplify the expression
Step 2: Solve for
Final Answer:
The value of is .
Let me know if you want further clarifications or steps! Here are five related questions to expand your understanding:
- What is the Factor Theorem, and how is it used in polynomial factorization?
- How can you check if a binomial is a factor of a given polynomial?
- If were a factor, what would the value of be?
- What is the connection between the remainder theorem and the factor theorem?
- Can you factorize the entire polynomial once ?
Tip: Always substitute the root back into the polynomial to verify your solution!
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomials
Factor Theorem
Formulas
Factor Theorem: If x - c is a factor of f(x), then f(c) = 0
Theorems
Factor Theorem
Suitable Grade Level
Grades 9-11