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Difficulty: HardMultiplying Polynomials
When analyzing algebraic interactions, it is often necessary to determine the exact coefficient of a specific term. Find the fully expanded and simplified form of the polynomial product (3a−4b)2(2a+b)(3a - 4b)^2(2a + b). What is the numerical coefficient of the a2ba^2b term?

Answer: -39

Answer

-39
To find the coefficient of the a2ba^2b term, first expand the squared binomial (3a−4b)2(3a - 4b)^2 to get the trinomial 9a2−24ab+16b29a^2 - 24ab + 16b^2. Next, multiply this trinomial by (2a+b)(2a + b). The specific terms that result in a2ba^2b are the product of 9a29a^2 and bb (which yields 9a2b9a^2b) and the product of −24ab-24ab and 2a2a (which yields −48a2b-48a^2b). Combining these like terms gives 9a2b−48a2b=−39a2b9a^2b - 48a^2b = -39a^2b. Therefore, the numerical coefficient is -39.

Step-by-Step Solution

1
Expand the squared binomial (3a−4b)2(3a - 4b)^2.
9a2−24ab+16b29a^2 - 24ab + 16b^2
Order of operations requires resolving the exponent before multiplying by the binomial (2a+b)(2a + b).
2
Set up the multiplication of the resulting trinomial and the binomial.
(9a2−24ab+16b2)(2a+b)(9a^2 - 24ab + 16b^2)(2a + b)
This prepares the expression for distributing every term in the trinomial to every term in the binomial.
3
Identify and calculate only the partial products that result in an a2ba^2b term.
(9a2)(b)=9a2b(9a^2)(b) = 9a^2b and (−24ab)(2a)=−48a2b(-24ab)(2a) = -48a^2b
This is a more efficient method than fully expanding the entire polynomial if only a specific coefficient is needed.
4
Combine the identified like terms.
9a2b−48a2b=−39a2b9a^2b - 48a^2b = -39a^2b
Combining these terms gives the final, simplified coefficient for the a2ba^2b term.

Key Concept

Multiplying polynomials, expanding binomial squares, and combining like terms.
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Topics

Multiplying Polynomials

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