Math Problem Statement

Simplify the following expression with distribution: (4p + 2 + q)(p^2 - 3q + 6)

Solution

Let's solve the given expression step by step using distribution.

The given expression is: (4p+2+q)(p23q+6)(4p + 2 + q)(p^2 - 3q + 6)

We will use the distributive property, multiplying each term from the first binomial by each term from the second binomial.

  1. Multiply 4p4p by each term in (p23q+6)(p^2 - 3q + 6):

4pp2=4p34p \cdot p^2 = 4p^3 4p(3q)=12pq4p \cdot (-3q) = -12pq 4p6=24p4p \cdot 6 = 24p

  1. Multiply 22 by each term in (p23q+6)(p^2 - 3q + 6):

2p2=2p22 \cdot p^2 = 2p^2 2(3q)=6q2 \cdot (-3q) = -6q 26=122 \cdot 6 = 12

  1. Multiply qq by each term in (p23q+6)(p^2 - 3q + 6):

qp2=p2qq \cdot p^2 = p^2q q(3q)=3q2q \cdot (-3q) = -3q^2 q6=6qq \cdot 6 = 6q

Now, combine all these terms together: 4p3+2p2+p2q12pq3q2+24p+(6q+6q)+124p^3 + 2p^2 + p^2q - 12pq - 3q^2 + 24p + (-6q + 6q) + 12

Simplifying, the terms involving qq cancel out, so we are left with: 4p3+2p2+p2q12pq3q2+24p+124p^3 + 2p^2 + p^2q - 12pq - 3q^2 + 24p + 12

This matches the third option in the image: 4p3+2p2+p2q12pq3q2+24p+12\boxed{4p^3 + 2p^2 + p^2q - 12pq - 3q^2 + 24p + 12}

Would you like more details on any step? Here are some additional questions to explore this further:

  1. How would the expression change if one of the terms was squared?
  2. Can this expression be factored after simplification?
  3. What happens if we change the order of multiplication (commutativity)?
  4. How would you approach this problem if it involved more variables?
  5. What is the importance of grouping like terms in simplifying polynomial expressions?

Tip: Always keep track of positive and negative signs when distributing across multiple terms to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Polynomial Multiplication
Distributive Property
Algebra

Formulas

(a+b)(c+d) = ac + ad + bc + bd

Theorems

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Suitable Grade Level

Grades 9-10