Question

if(x-2) and (x+3) are factors of x³+ax²+bx-30 find a and b.​

Answer 1

$\huge\red{QUESTION}$

$(x - 2) \: and \: (x + 3) \: are \: factors \: of \: {x}^{3} + a {x}^{2} + bx - 30$

$\huge\mathfrak\blue{TO FIND}$

$a \: and \: b$

$x - 2 = 0 \\ (as \: it \: is \: the \: factor \: of \: polynomial)$

$x = 2$

$x + 3 = 0 \\ x = - 3$

PUTTING THE BOTH VALUES ONE BY ONE IN GIVEN POLYNOMIAL.

PUTTING 2 in polynomial

$p(x) \: = {x}^{3} + a {x}^{2} + bx - 30 \\ p(2) \: = {2}^{3} + a( {2})^{2} + b \times 2 - 30 = 0 \\ p(2) = 8 + 4a + 2b - 30 = 0 \\ p(2) = 4a + 2b = 22 \\ 2(2a + b) = 22 \\ 2a + b = 11..........(eq.1)$

putting -3 in polynomial

$p( - 3) = { - 3}^{3} + a {( - 3)}^{2} + b( - 3) - 30 = 0 \\ - 27 + 9a - 3b - 30 = 0 \\ - 57 + 9a - 3b = 0 \\ 9a - 3b = 57 \\ 3(3a + b) = 57 \\ 3a + b = 19............(eq.2)$

subtracting both the equation:

2a+b=11

3a+b=19

(-) (-). (-)

_______

-a = 8

a = -8

$putting \: the \: value \: of \: a \: in \: eq.1$

$2a + b = 11 \\ 2( - 8) + b = 11 \\ - 16 + b = 11 \\ b = 27$

so final ANSWER:

$\huge\mathbb\red{a = -8;</p><p>b= 27}$