https://file.aiquickdraw.com/mathbot/file/1726781578581k957x3ns.jpg

The given polynomial function is:

\[
p(x) = x^3 + 3x^2 - 6x - 8
\]

and it is stated that \((x + 4)\) is a known factor of \(p(x)\).

To rewrite \(p(x)\) as a product of linear factors, we need to divide \(p(x)\) by \((x + 4)\) using either synthetic division or long division.

Let's proceed with synthetic division to factor \(p(x)\).

1. The coefficients of the polynomial \(p(x)\) are: \(1\) (for \(x^3\)), \(3\) (for \(x^2\)), \(-6\) (for \(x\)), and \(-8\) (the constant term).
2. Since \(x + 4\) is a factor, we will use \(x = -4\) in synthetic division.

I'll calculate this next.The result of the synthetic division is:

\[
p(x) = (x + 4)(x^2 - x - 2)
\]

Now, to fully factor the quadratic part \(x^2 - x - 2\), we can factor it into linear factors:

\[
x^2 - x - 2 = (x - 2)(x + 1)
\]

Thus, the polynomial \(p(x)\) can be written as the product of linear factors:

\[
p(x) = (x + 4)(x - 2)(x + 1)
\]

Would you like a more detailed explanation of any step?

Here are 5 related questions to consider:
1. How do you apply synthetic division for polynomials?
2. Why is \(x = -4\) used for synthetic division with \(x + 4\)?
3. What are other methods for factoring quadratic expressions?
4. Can synthetic division be used if a factor isn't provided?
5. How do you verify if a given expression is a factor of a polynomial?

**Tip:** Always check your factorization by expanding the factors to ensure you retrieve the original polynomial.

The polynomial function p(x) = x^3 + 3x^2 - 6x - 8 has a known factor of (x + 4). Rewrite p(x) as the product of linear factors.

Learn how to rewrite the cubic polynomial p(x) = x^3 + 3x^2 - 6x - 8 as a product of linear factors using synthetic division. Understand key algebraic concepts like the Factor Theorem and polynomial division, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation