Question

factorize using the factor theorem 2x^3− 13x^2+6x+45

Answer 1

$\huge{\boxed{\red{\bf{Solution:-}}}}$

Let ${f(x)} = {2x}^{3} - {13x}^{2} + {6x} + {45}$

then,

let ${x} = {3}$

${f(3)} = {2×3}^{3} - {13×3}^{2} + {6×3} + {45}$

${f(3)} = {54} - {117} + {18} + {45}$

${f(3)} = {117} - {117}$

${f(3)} = {0}$

(x-3) is a factor of f(x).

Dividing f(x) from (x+1), we get:-

${2x}^{2} - {7x} - {15}={0}$

Factorizing ${2x}^{2} - {7x} - {15}={0}$:-

${2x}^{2} - {10x} + {3x} - {15}={0}$

${2x(x-5)-3(x-5)}={0}$

${(2x-3)(x-5)}={0}$

$\huge{\boxed{\red{\bf{Answer:-}}}}$

Hence the factors of f(x) are:-

(x - 5)(2x - 3)(x - 3)