Question

Given that the expression 2x³+px²-8x+q is exactly divisible by 2x²-7x+6 evaluate p and q and factorize the expression fully

Answer 1

$\textbf{ Hello Mate! }$

The polynomial given here,

$f(x) = 2 {x}^{3} + p {x}^{2} - 8x + q$

The hiven factor is,

$2 {x}^{2} - 7x + 6 = 0 \\ 2 {x}^{2} - 4x - 3x + 6 = 0 \\ 2x(x - 2) - 3(x - 2) = 0 \\ (2x - 3)(x - 2) = 0 \\ \: each \: getting \: 0 \\ (x - 2) = 0 \\ x = 2 \\ \\ 2x - 3 = 0 \\ x = \frac{3}{2}$

$f(2) = 2 {(2)}^{3} + p {(2)}^{2} - 8(2) + q \\ 0 = 16 + 4p - 16 + q \\ 0 = 4p + q......(1)$

$f( \frac{3}{2} ) = 2 {( \frac{3}{2} )}^{3} + p {( \frac{3}{2} )}^{2} - 8( \frac{3}{2} ) + q \\ 0 = \frac{27}{4} + \frac{9p}{4} - 9 + q \\ 0 = \frac{27 + 9p - 36 + 4q}{4} \\ 0 \times 4 = - 9 + 9p + 4q \\ 9 = 9p + 4q.......(2)$

Here, we get two equations.

Multiplying -4 from equation 1 we get

0 = - 16p - 4q .......(3)

Now adding equation (2) and (3) we get

9 = - 7p

- 9 / 7 = p

Keeping value in equation (1)

0 = 4 × ( - 9 / 7 ) + q

0 = - 36 / 7 + q

36 / 7 = q

$\textbf{ Have great future ahead! }$