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SAT Math : How to find the solution to a binomial problem

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Example Question #1 : Binomials

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real m, n ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Possible Answers:

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

$m \heartsuit n = m ^{3} +12 m ^{2} n - 48mn ^{2}-64n ^{3}$

$m \heartsuit n = m ^{3} -4 m ^{2} n + 16mn ^{2}- 64n ^{3}$

$m \heartsuit n = m ^{3} +4 m ^{2} n - 16mn ^{2}- 64n ^{3}$

$m \heartsuit n = m ^{3} -64n ^{3}$

Correct answer:

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Explanation:

$\left ( m-4n\right )^{3}$ , as the cube of a binomial, can be rewritten using the following pattern:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot m ^{2} \cdot 4n + 3 \cdot m \cdot (4n) ^{2}- (4n) ^{3}$

Applying the rules of exponents to simplify this:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot m \cdot 4 ^{2}\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot 16 \cdot m\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Therefore, the correct choice is that, alternatively stated,

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$ .

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SAT Math : How to find the solution to a binomial problem

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Example Question #1 : Binomials

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