How to multiply polynomials

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PSAT Math › How to multiply polynomials

Questions 1 - 2

1

Multiply:

$\left (100Y^{2}- 70Y + 49 \right )(10Y+7)$

$1,000Y^{3} + 343$

$1,000Y^{3} + 700Y^{2}- 490 Y- 343$

$1,000Y^{3} - 700Y^{2}+ 490 Y- 343$

$1,000Y^{3} - 100Y^{2}+49 Y- 343$

$1,000Y^{3}+ 100Y^{2}-49 Y- 343$

Explanation

$\left (100Y^{2}- 70Y + 49 \right )(10Y+7)$

$= \left [\left (10Y \right )^{2}- 10Y\cdot 7 +7^{2} \right ](10Y+7)$

This product fits the sum of cubes pattern, where :

$(A^{2}-AB +B^{2})(A+B) = A^{3}+B^{3}$

So

$\left [\left (10Y \right )^{2}- 10Y\cdot 7 +7^{2} \right ](10Y+7)$

$= (10Y)^{3}+7^{3} = 1,000Y^{3} + 343$

2

$F(x) = x^{3} + x^{2} - x + 2$

and

$G(x) = x^{2} + 5$

What is ?

$(FG)(x) = x^{5} + x^{4} +4x^{3} + 7x^{2} - 5x +10$

$(FG)(x) = x^{5} + x^{4} - x^{3} + 2x^{2} - 5x -10$

$(FG)(x) = x^{5} + x^{4} - x - 2$

$(FG)(x) = x^{3} + 2x^{2} - x + 7$

$(FG)(x) = x^{3} - x - 3$

Explanation

$(FG)(x) = F(x)G(x)$ so we multiply the two function to get the answer. We use $x^{m}x^{n} = x^{m+n}$

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