Question

show that 2x+7 is a factor of 2x^3+5x^2-11x-14.hence factorise the given expression completely, using the factor theorem​

Answer 1

I have attached the answer

Answer 2

Solution :-

we know that, if (x + a) is a factor of any polynomial p(x) , than, at p(-a) , value of p(x) is equal to zero.

So,

→ 2x + 7 = 0

→ 2x = (-7)

→ x = (-7/2)

Putting x = (-7/2), we gets,

→ p(x) = 2x^3 + 5x^2 - 11x - 14

→ p(-7/2) = 2(-7/2)³ + 5(-7/2)² - 11(-7/2) - 14

→ p(-7/2) = 2(-343/8) + 5(49/4) + (77/2) - 14

→ p(-7/2) = (-686/8) + (245/4) + (77/2) - (14/1)

→ p(-7/2) = (-686 + 490 + 308 - 112)/8

→ p(-7/2) = (798 - 798) / 8

→ p(-7/2) = 0 .

Therefore, (2x + 7) is a factor of 2x^3+5x^2-11x-14 .

Now, Dividing the given polynomial by (2x + 7) , we get,

2x+7 )2x³ + 5x² - 11x - 14(x² - x - 2

-2x³ + 7x²

-2x² - 11x

-2x² - 7x

-4x - 14

-4x - 14

__0___

Quotient = (x² - x - 2)

using common factor theorem we get,

→ x² - 2x + x - 2

→ x (x - 2) + 1(x - 2)

→ (x - 2)(x + 1)

Hence, the factorise form of the given polynomial is (x + 2)(x - 2)(2x + 7) .