What kind of special product should i use to multiply this: 

(w²-7wv+u²+v²)(7wv+v²+w²+u²)

To multiply the given expression \((w^2 - 7wv + u^2 + v^2)(7wv + v^2 + w^2 + u^2)\), you would use **distribution**, or the **FOIL method**, if there are fewer terms. However, since these are binomials with four terms each, this doesn't directly fit a standard **special product** formula (like perfect squares, difference of squares, etc.).

Instead, you can:

1. Distribute each term in the first expression \((w^2 - 7wv + u^2 + v^2)\) to each term in the second expression \((7wv + v^2 + w^2 + u^2)\).
2. Combine like terms, if any, after multiplying.

### Step-by-step:

Let's break it down:

1. **First term in the first binomial**: Multiply \( w^2 \) by each term in the second binomial.
   \[
   w^2(7wv + v^2 + w^2 + u^2) = 7w^3v + w^2v^2 + w^4 + w^2u^2
   \]

2. **Second term in the first binomial**: Multiply \( -7wv \) by each term in the second binomial.
   \[
   -7wv(7wv + v^2 + w^2 + u^2) = -49w^2v^2 - 7wv^3 - 7w^3v - 7wu^2v
   \]

3. **Third term in the first binomial**: Multiply \( u^2 \) by each term in the second binomial.
   \[
   u^2(7wv + v^2 + w^2 + u^2) = 7wu^2v + u^2v^2 + u^2w^2 + u^4
   \]

4. **Fourth term in the first binomial**: Multiply \( v^2 \) by each term in the second binomial.
   \[
   v^2(7wv + v^2 + w^2 + u^2) = 7wv^3 + v^4 + v^2w^2 + v^2u^2
   \]

### Combine all the terms:

\[
7w^3v + w^2v^2 + w^4 + w^2u^2 - 49w^2v^2 - 7wv^3 - 7w^3v - 7wu^2v + 7wu^2v + u^2v^2 + u^2w^2 + u^4 + 7wv^3 + v^4 + v^2w^2 + v^2u^2
\]

Now combine like terms:

- \( 7w^3v - 7w^3v = 0 \)
- \( w^2v^2 - 49w^2v^2 + v^2w^2 = -47w^2v^2 + v^2w^2 = -46w^2v^2 \)
- \( w^4 \)
- \( w^2u^2 + u^2w^2 \)
- \( -7wv^3 + 7wv^3 = 0 \)
- \( -7wu^2v + 7wu^2v = 0 \)
- \( u^2v^2 + v^2u^2 = 2u^2v^2 \)
- \( u^4 \)
- \( v^4 \)

Thus, the final expression becomes:

\[
w^4 + u^4 + v^4 + 2w^2u^2 - 46w^2v^2 + 2u^2v^2
\]

This is the simplified result of the multiplication.

---

Do you have any questions or would you like further details?

---

Here are 5 related questions for practice:

1. What is the result of \((x^2 + y^2)(x^2 - y^2)\)?
2. How do you simplify \((a+b+c)(a-b+c)\)?
3. What are the steps for multiplying two trinomials?
4. Can you apply any special product rules to \((w^2 + u^2)(w^2 + u^2)\)?
5. How do you expand \((w+x+y+z)^2\)?

**Tip:** Always look for opportunities to factor common terms when multiplying polynomials!

What kind of special product should I use to multiply this: (w² - 7wv + u² + v²)(7wv + v² + w² + u²)?

Learn how to multiply two polynomials (w² - 7wv + u² + v²)(7wv + v² + w² + u²) using distribution and the distributive property. This step-by-step guide shows how to break down each term and combine like terms to arrive at the simplified expression, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation