Question

Factorise 6x³-25x²+32x-12​

Answer 1

Answer:

(x-2)(-3x+2)(-2x+3)

Step-by-step explanation:

6x³-25x²+32x-12

6×12=72     (First term×Last term)

Factors of 72=±1,±2,±3,±4,±5,±6,±8,±9,±12,±18,24,±36±72

P(x)=6x³-25x²+32x-12     (Multiply with each factor)

P(1)=6(1)³-25(1)²+32(x)-12

     =6-25+32-12

     =1

P(-1)=6(-1)³-25(-1)²+32(-1)-12     (Multiply till the product is 0)

      =-6-25-32-12

      =-31

P(2)=6(2)³-25(2)³+32(2)-12

     =6(8)-25(4)+32(2)-12

     =48-100+64-12

     =0

(x-2) is the factor of p(x).

Take the factor with which when multiplied got the product 0 as the divisor and the co-effecients of p(x) as dividend for synthetic division.

2|6-25+32-12

 |    12  -26 12

 |__________

  6 -13    6   0

Change the quotient into quadratic polynomial and factorise.

Q(x)=6x²-13x+6                    (First term ×last term=6×6=36)

      =6x²-9x-4x+6               (Factors of 36=1×36,2×18,3×12,4×9,6×6)                                                                                                  

      =-3x²(-2x+3)+2(-2x+3)

      =(-3x+2)(-2x+3)

(x-2),(-3x+2),(-2x+3) are the factors of P(x).

Answer 2

$\textbf{\huge\underline{\underline\red{Solution} : - }} \\ \\ \bold{ \underline\red{let }} \\ \\\bold{\purple {f(x ) =6{x}^{3} - 25 {x}^{2} + 32x - 12 }} \\ \\ \bold{ \underline\red {now \: take \: x \: = \: 2}} \\ \bold{ \underline\red{so}} \\ \\\bold{\purple {f(x = 2) =6{(2)}^{3} - 25 {(2)}^{2} + 32(2) - 12 }} \\ \\\bold{\purple {f(x = 2) =6(8) - 25 (4) + 64 - 12 }} \\ \\\bold{\purple {f(x = 2) =48 - 100 + 52 }} \\ \\\bold{\purple {f(x = 2) = - 52 + 52 }} \\ \\ \bold{\purple {f(x = 2) = 0 }} \\ \\ \bold{ \underline\blue{on \: puting \: x \: = \: 2 \: \:we \: get \: f(x) = \: 0} }\\ \bold{ \underline\blue{hence \: one \: of \: the \: factor \: of \: given \: }} \\ \bold{ \underline\blue{polynomial \: is \: (x - 2).}} \\ \\ \bold{\underline \red{now}}\\ \\\bold{\purple { =>6{x}^{3} - 25 {x}^{2} + 32x - 12 }} \\ \\ \bold{\purple { =( x - 2)(6 {x}^{2} - 13x + 6 )}} \\ \\ \bold{\purple {=( x - 2)(6 {x}^{2} - 9x - 4x + 6 )}} \\ \\ \bold{\purple { =( x - 2)(3x(2x - 3) - 2( 2x - 3))}} \\ \\ \bold{\purple { =( x - 2)(2x- 3)( 3x - 2)}} \\ \\ \fbox{\bold{ \underline\red{i \: hope \: it \: helps \: you.}}}$