For all real numbers m and n, the expression 4m(2m - 3n)^2 - n(5m^2 - 36mn) can be written… — ACT

For all real numbers $m$ and $n$ , the expression $4m(2m - 3n)^2 - n(5m^2 - 36mn)$ can be written in the form $am^3 + bm^2n + cmn^2$ , where $a$ , $b$ , and $c$ are real constants. What is the value of the coefficient $b$ ?

Answer: -53

Answer

The value of the coefficient

b

-53

Expanding the entire expression yields

16m^3 - 53m^2n + 72mn^2

. Comparing this to the template form

am^3 + bm^2n + cmn^2

shows that

b = -53

Step-by-Step Solution

Expand the squared binomial

(2m - 3n)^2

4m^2 - 12mn + 9n^2

Apply the identity

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

to expand the expression inside the parentheses.

Distribute

4m

through the trinomial

16m^3 - 48m^2n + 36mn^2

Multiply each term of

4m^2 - 12mn + 9n^2

4m

using properties of exponents.

Distribute

-n

across

(5m^2 - 36mn)

-5m^2n + 36mn^2

Multiply

-n

by both terms inside the parentheses, paying attention to sign rules.

Group and combine the like terms

16m^3 + (-48m^2n - 5m^2n) + (36mn^2 + 36mn^2)

Identify terms with the same variables and exponents to combine them.

Combine the coefficients of the like terms

16m^3 - 53m^2n + 72mn^2

Perform arithmetic on the coefficients:

-48 - 5 = -53

for the

m^2n

terms and

36 + 36 = 72

for the

mn^2

terms.

Identify the coefficient

b

-53

Compare the simplified polynomial to the form

am^3 + bm^2n + cmn^2

to find the value of

b

Key Concept

Simplifying algebraic expressions by expanding binomials, distributing variables and signs, and combining like terms.

Alternative Method

Instead of expanding the entire expression, focus only on terms that produce

m^2n

: from the first part,

4m \times (-12mn) = -48m^2n

, and from the second part,

-n \times 5m^2 = -5m^2n

. Adding these gives

-53m^2n

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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