The expression 2x3+ax2+bx-2 is divisible by x+2 but leaves reamainder 7 when divide by 2x-3 Determine values of a - Maths - Polynomials

Question

The expression 2x3+ax2+bx-2 is divisible
by x+2 but leaves reamainder 7 when divide by 2x-3 Determine values
of a & b

Ajanta Trivedi · Accepted Answer

let the given polynomial be $p (x) = 2 x^{3} + a x^{2} + b x - 2$ .

since p(x) is divisible by (x+2), put x+2=0 ⇒x = -2

therefore

$p (- 2) = 0 2 * (- 2)^{3} + a (- 2)^{2} + b * (- 2) - 2 = 0 - 16 + 4 a - 2 b - 2 = 0 4 a - 2 b = 18 2 a - b = 9.......... (1)$

when p(x) is divided by (2x-3), remainder = 7.

put 2x-3 = 0 ⇒x = 3/2

$p (\frac{3}{2}) = 7 2 * (\frac{3}{2})^{3} + a (\frac{3}{2})^{2} + b * (2) - 2 = 7 2 * (\frac{27}{8}) + a * \frac{9}{4} + 2 b - 2 = 7 \frac{27}{4} + \frac{9}{4} a - 2 b - 2 = 7$

$\frac{9}{4} a - 2 b = 2 + 7 - \frac{27}{4} \frac{9}{4} a - 2 b = 9 - \frac{27}{4} = \frac{36 - 27}{4} \frac{9 a}{4} - 2 b = \frac{9}{4} 9 a - 8 b = 9........... (2)$

multiplying eq(1) by 8 and subtracting eq(2):

$16 a - 8 b = 72 9 a - 8 b = 9 7 a = 63 a = \frac{63}{7} = 9$

substitute the value of a in eq(1):

$2 * 9 - b = 9 b = 18 - 9 b = 9$

hope this helps you.

cheers!!

The expression 2x3+ax2+bx-2 is divisible by x+2 but leaves reamainder 7 when divide by 2x-3.Determine values of a & b.

The expression 2x³+ax²+bx-2 is divisible by x+2 but leaves reamainder 7 when divide by 2x-3.Determine values of a & b.