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 KK such that the binomial x3x - 3 is a factor of the cubic polynomial f(x)=x3+Kx2+x+6f(x) = x^3 + Kx^2 + x + 6, we use the Factor Theorem.

The Factor Theorem states: If xcx - c is a factor of f(x)f(x), then f(c)=0f(c) = 0.

Here, c=3c = 3 because the factor is x3x - 3. Substituting x=3x = 3 into f(x)f(x), we get: f(3)=33+K(32)+3+6=0f(3) = 3^3 + K(3^2) + 3 + 6 = 0

Step 1: Simplify the expression

f(3)=27+9K+3+6=0f(3) = 27 + 9K + 3 + 6 = 0

27+3+6+9K=027 + 3 + 6 + 9K = 0

36+9K=036 + 9K = 0

Step 2: Solve for KK

9K=369K = -36

K=4K = -4

Final Answer:

The value of KK is K=4K = -4.

Let me know if you want further clarifications or steps! Here are five related questions to expand your understanding:

  1. What is the Factor Theorem, and how is it used in polynomial factorization?
  2. How can you check if a binomial is a factor of a given polynomial?
  3. If x+2x + 2 were a factor, what would the value of KK be?
  4. What is the connection between the remainder theorem and the factor theorem?
  5. Can you factorize the entire polynomial f(x)f(x) once K=4K = -4?

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