Factorise $9 a^{3} - 27 a^{2} - 100 a + 300$ , if $3 a + 10$ is a factor of it.

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Solution

Here given that $3 a + 10$ is a factor os
$f (a) = 9 a^{3} - 27 a^{2} - 100 a + 300$
Now, dividing the polynomial by division process

${3 a}^{2} - 19 a + 30 3 a + 10) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ {9 a}^{3} - 27 a^{2} - 100 a + 300 {9 a}^{3} + 30 a^{2}_________- {57 a}^{2} - 100 a {- 57 a}^{2} - 190 a_90 a + 300 90 a + 300_________0$

Therefore Quotient: $3 a^{2} - 19 a + 30$
Remainder $0$
$∴ 9 a^{3} - 27 a^{2} - 100 a + 300 = (3 a + 10) (3 a^{2} - 19 a + 30)$
$\Rightarrow$ Factorizing the quadratic polynomial,
$3 a^{2} - 19 a + 30 = 3 a^{2} - 10 a - 9 a + 30$
$= a (3 a - 10) - 3 (3 a - 10)$
$= (a - 3) (3 a - 10)$
$\to$ Thus factors of $9 a^{3} - 27 a^{2} - 100 a + 300$ are:
$(3 a + 10), (a - 3) (3 a - 10)$

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