Math Problem Statement

Expand and simplify the expression (3a^2 + 2ab + c)(3c - 4a^2 - ab).

Solution

The expression you provided is:

(3a2+2ab+c)(3c4a2ab)(3a^2 + 2ab + c)(3c - 4a^2 - ab)

To solve this, we will expand the expression by applying the distributive property, also known as the FOIL method (First, Outer, Inner, Last) for binomials, and fully distribute each term in the first parentheses by each term in the second parentheses.

Let's go through the steps:

  1. Multiply 3a23a^2 by each term in the second parentheses:

    • 3a23c=9a2c3a^2 \cdot 3c = 9a^2c
    • 3a2(4a2)=12a43a^2 \cdot (-4a^2) = -12a^4
    • 3a2(ab)=3a3b3a^2 \cdot (-ab) = -3a^3b
  2. Multiply 2ab2ab by each term in the second parentheses:

    • 2ab3c=6abc2ab \cdot 3c = 6abc
    • 2ab(4a2)=8a3b2ab \cdot (-4a^2) = -8a^3b
    • 2ab(ab)=2a2b22ab \cdot (-ab) = -2a^2b^2
  3. Multiply cc by each term in the second parentheses:

    • c3c=3c2c \cdot 3c = 3c^2
    • c(4a2)=4a2cc \cdot (-4a^2) = -4a^2c
    • c(ab)=abcc \cdot (-ab) = -abc

Now we combine all these terms:

9a2c12a43a3b+6abc8a3b2a2b2+3c24a2cabc9a^2c - 12a^4 - 3a^3b + 6abc - 8a^3b - 2a^2b^2 + 3c^2 - 4a^2c - abc

Next, we’ll group and simplify like terms:

  • The a2ca^2c terms: 9a2c4a2c=5a2c9a^2c - 4a^2c = 5a^2c
  • The abcabc terms: 6abcabc=5abc6abc - abc = 5abc
  • The a3ba^3b terms: 3a3b8a3b=11a3b-3a^3b - 8a^3b = -11a^3b

So the simplified expression is:

12a4+5a2c11a3b2a2b2+3c2+5abc-12a^4 + 5a^2c - 11a^3b - 2a^2b^2 + 3c^2 + 5abc

Would you like more details on any specific step, or do you have further questions?

Related Questions

  1. How do you apply the distributive property to polynomial expressions?
  2. What are some strategies for simplifying expressions with multiple terms?
  3. How do you identify like terms in polynomial expressions?
  4. What is the FOIL method and when is it useful?
  5. How does combining like terms help in simplifying algebraic expressions?

Tip

When expanding expressions with multiple variables, it helps to systematically distribute each term to avoid missing any components.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Expansion
Distributive Property

Formulas

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

Theorems

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Suitable Grade Level

Grades 9-10