If x + 3 is a factor of x3 + 3x2 + ax - 12, what is the value of a?

We are given that \( x + 3 \) is a factor of the polynomial \( x^3 + 3x^2 + ax - 12 \). To find the value of \( a \), we can use the **Factor Theorem**, which tells us that if \( x + 3 \) is a factor, then \( x = -3 \) should satisfy the polynomial.

Let's substitute \( x = -3 \) into the polynomial and solve for \( a \).

The polynomial is:
\[
P(x) = x^3 + 3x^2 + ax - 12
\]

Substitute \( x = -3 \):
\[
(-3)^3 + 3(-3)^2 + a(-3) - 12 = 0
\]
Simplify:
\[
-27 + 3(9) + (-3a) - 12 = 0
\]
\[
-27 + 27 - 3a - 12 = 0
\]
\[
-12 - 3a = 0
\]
Now, solve for \( a \):
\[
-3a = 12
\]
\[
a = -4
\]

Thus, the value of \( a \) is \( \boxed{-4} \).

Would you like more details or explanations on this? Here are 5 related questions you could explore:
1. How can the Factor Theorem help in factoring polynomials?
2. What is synthetic division, and how could it be used here?
3. How do you find other factors of a cubic polynomial?
4. What are the roots of the polynomial after factoring?
5. Can we verify \( a = -4 \) by expanding and comparing terms?

**Tip:** The Factor Theorem is a powerful tool in simplifying polynomials by testing for possible roots based on factors.

If x + 3 is a factor of x^3 + 3x^2 + ax - 12, what is the value of a?

Learn how to find the value of 'a' in the polynomial x^3 + 3x^2 + ax - 12 using the Factor Theorem. This problem explores key algebraic concepts like factoring polynomials and applying the Factor Theorem, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation