Question

Factorise -:
6x²+5x-6 by factor theorem.​

Answer 1

Mark me as BRAINIST vote me and thank me

Answer 2

$\huge{ \red{ꕤ}}\huge{\color{blue}{ \boxed{ \color{yellow}{ \overbrace{ \underbrace{ \fcolorbox{red}{orange}{\pink{\underline{\ttภnşwer}}}}}}}}}\: \: \huge{ \red{꧁}}$

Given-

$\large6 {x}^{2} + 5x - 6$

To factorize using factor theorem.

Solution-

$\large \implies 6 {x}^{2} + 5x - 6$

(Taking 6 common)

$\large \implies 6 \huge( \large {x}^{2} + \huge \frac{5x}{6} \large - 1 \huge)$

$\large Let \: f(x) = {x}^{2} + \huge \frac{5x}{6} \large - 1$

Then,

$\large \implies 6 \huge( \large {x}^{2} + \huge \frac{5x}{6} \large - 1 \huge) \large = 6 \times f(x)$

Now,

By trial method, we get:

$\large f \huge(\frac{ - 3}{2} ) \large = \huge{( \frac{ - 3}{2} ) }^{2} \large + \huge \frac{5}{6} \large \times \huge (\frac{ - 3}{2} ) \large - 1$

$\large = \huge \frac{9}{4} \large - \huge \frac{15}{12} \large - 1$

$\large = \huge \frac{27 - 15 - 12}{12}$

$\large = \huge \frac{0}{12}$

$\large = 0$

Hence, by factor theorem,

$\huge( \large x + \huge \frac{3}{2} ) \large \: is \: a \: factor \: of \: f(x).$

Now, by long division method,

$\large x + \frac{3}{2} \: \overline{) {x}^{2} + \frac{5x}{6} - 1( } x - \frac{4}{6} \\ \large \underline{{ _{ - }x}^{2} + _{ - }\frac{3x}{2} } \\ \large \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \frac{ - 4x}{6} - 1 \\ \large \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ _{ + } \frac{ - 4x}{6} - _{ + } 1} \\ \large \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 0}$

Hence,

$\huge (x - \frac{4}{6})\large \:is\:a\:factor\:of\:f(x).$

Therefore,

$\large6 \times f(x) = 6( {x}^{2} + \huge \frac{5x}{6} \large - 1)$

$\large = 6(x + \huge \frac{3}{2} \large )(x - \huge \frac{ \cancel4}{ \cancel6} \large)$

$\large = \cancel6\huge (\frac{2x + 3}{ \cancel2} )( \frac{3x - 2}{ \cancel3} )$

$\large =( 2x + 3)(3x - 2)$

Final Answer-

$\large \green{ \boxed{ \bold{6 {x}^{2} + 5x - 6 = (2x + 3)(3x - 2)}}}$

Note-

Middle term splitting method can also be used to get the answer.

$\large{ \pink{ \mathsf{ \underline{ \underline{Additional \: Information-}}}}}$

● Remainder Theorem:

Let f(x) be the given polynomial of degree more than or equal to 1 and 'a' is any real number. If f(x) is divided by (x - a), then remainder = f(a).

● Factor Theorem:

Let f(x) be the given polynomial of degree more than or equal to 1 and 'a' is any real number, then-

(i) If f(a) = 0, then (x - a) is a factor of f(x).

(ii) If (x - a) is a factor of f(x), then f(a) = 0.