Binomials - SAT Math

Card 0 of 3

Question

Solve for .

$\frac{x^{2}+2x-8}{x^{2}+5x-14}=\frac{3}{4}$

Answer

$\frac{x^{2}+2x-8}{x^{2}+5x-14}=\frac{3}{4}$

Factor the expression

numerator: find two numbers that add to 2 and multiply to -8 \[use 4,-2\]

denominator: find two numbers that add to 5 and multiply to -14 \[use 7,-2\]

new expression:

$\frac{(x+4)(x-2)}{(x+7)(x-2)}=\frac{3}{4}$

Cancel the and cross multiply.

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Question

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

For all real ,

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

How else could this operation be defined?

Answer

$\left ( m-4n\right )^{3}$ , 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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Question

If 〖(x+y)〗2 = 144 and 〖(x-y)〗2 = 64, what is the value of xy?

Answer

We first expand each binomial to get x2 + 2xy + y2 = 144 and x2 - 2xy + y2 = 64. We then subtract the second equation from the first to find 4xy = 80. Finally, we divide each side by 4 to find xy = 20.

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