Is $(3 x^{2} + 5 x y + 4 y^{2})$ a factor of $9 x^{4} + 3 x^{3} y + 16 x^{2} y^{2} + 24 x y^{3} + 32 y^{4}$ ?

True

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False

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Solution

The correct option is A True
$3 x^{2} + 5 x y + 4 y^{2}$ $¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯) 9 x^{4} + 3 x^{3} y + 16 x^{2} y^{2} + 24 x y^{3} + 32 y^{4} ($ $3 x^{2} - 4 x y + 8 y^{2}$
$- (3 x^{4} + 15 x^{3} y + 12 x^{2} y^{2})$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ - 12 x^{3} y + 4 x^{2} y^{2} + 24 x y^{3}$
$- (- 12 x^{3} y - 20 x^{2} y^{2} - 16 x y^{3})$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 24 x^{2} y^{2} + 40 x y^{3} + 32 y^{4}$
$- (24 x^{2} y^{2} + 40 x y^{3} + 32 y^{4})$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 0$
As, the remainder is 0, so, $3 x^{2} + 5 x y + 4 y^{2}$ is a factor of $9 x^{4} + 3 x^{3} y + 16 x^{2} y^{2} + 24 x y^{3} + 32 y^{4}$
so, TRUE

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