Consider the following algebraic expression: (2x - 5y)^2(x + 2y) If you were to fully expa… — Armed Services Vocational Aptitude Battery (ASVAB)

Consider the following algebraic expression:

(2x - 5y)^2(x + 2y)

If you were to fully expand this expression and combine all like terms, what would be the numerical coefficient of the $x^2y$ term?

Answer:The numerical coefficient of the

x^2y

term is 【-12】.

-12

The correct answer is derived by first expanding the squared binomial:

(2x - 5y)^2 = 4x^2 - 20xy + 25y^2

. Next, multiply this resulting trinomial by the remaining binomial:

(4x^2 - 20xy + 25y^2)(x + 2y)

. To find the coefficient of the

x^2y

term without expanding the entire expression, identify the term pairs whose product yields

x^2y

. These pairs are

(4x^2)(2y) = 8x^2y

and

(-20xy)(x) = -20x^2y

. Combining these like terms gives

8x^2y - 20x^2y = -12x^2y

. Thus, the numerical coefficient of the

x^2y

term is

-12

Expand the squared binomial

(2x - 5y)^2

(2x)^2 - 2(2x)(5y) + (5y)^2 = 4x^2 - 20xy + 25y^2

Order of operations requires evaluating exponents before multiplying by the binomial

(x + 2y)

Set up the multiplication of the resulting trinomial and the remaining binomial.

(4x^2 - 20xy + 25y^2)(x + 2y)

Substitute the expanded form back into the original expression to continue the multiplication process.

Identify only the specific multiplications that will produce an

x^2y

term.

The term

4x^2

multiplied by

2y

, and the term

-20xy

multiplied by

x

To find the coefficient of a specific term, it is most efficient to calculate only the products that result in that exact variable combination.

Calculate and combine the identified products to find the final coefficient.

(4x^2)(2y) + (-20xy)(x) = 8x^2y - 20x^2y = -12x^2y

. The numerical coefficient is

-12

Combining these like terms gives the total

x^2y

term in the fully expanded polynomial.

Multiplying Polynomials

Question