(p²-4) (p-3) (p+6) The sum of its producers

The given expression is \((p^2 - 4)(p - 3)(p + 6)\). To find the sum of the products of the factors, you need to expand the expression fully and then add the coefficients of the resulting polynomial.

### Step 1: Factorize and Expand
First, notice that \(p^2 - 4\) can be factorized as \((p - 2)(p + 2)\). So the expression becomes:
\[
(p - 2)(p + 2)(p - 3)(p + 6)
\]

### Step 2: Multiply the First Two Binomials
Multiply \((p - 2)\) and \((p + 2)\):
\[
(p - 2)(p + 2) = p^2 - 4
\]

### Step 3: Multiply the Result with the Next Binomial
Now multiply \(p^2 - 4\) by \((p - 3)\):
\[
(p^2 - 4)(p - 3) = p^3 - 3p^2 - 4p + 12
\]

### Step 4: Multiply by the Last Binomial
Now, multiply the resulting expression by \((p + 6)\):
\[
(p^3 - 3p^2 - 4p + 12)(p + 6)
\]
Expanding this:
\[
= p^4 + 6p^3 - 3p^3 - 18p^2 - 4p^2 - 24p + 12p + 72
\]
\[
= p^4 + 3p^3 - 22p^2 - 12p + 72
\]

### Step 5: Sum of the Coefficients
The sum of the products of the coefficients is the sum of the coefficients of the polynomial:
\[
1 + 3 - 22 - 12 + 72 = 42
\]

Thus, the sum of the products of the factors is **42**.

Would you like any more details, or do you have any questions?

Here are 5 related questions:
1. How do you factorize quadratic expressions like \(p^2 - 4\)?
2. What is the process to multiply binomials?
3. Can you explain the steps to expand polynomials?
4. How do you determine the sum of the coefficients of a polynomial?
5. What are the applications of polynomial factorization?

**Tip:** When expanding polynomials, always multiply in small steps to avoid errors.

Explore the factorization and expansion of the expression (p²-4)(p-3)(p+6) step-by-step. Learn how to multiply binomials and expand polynomials effectively. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation