Question

19.If x² + x 12 divides P(x) = x + ax² + bx - 84 exactly, find a and b.​

Answer 1

Answer:

Step-by-step explanation:

Correct Question :-

If x² + x - 12 divides f(x) = x³ + ax² + bx - 84 exactly. find a and b.

Given :-

The equation  x² + x - 12 divides the equation x³ + ax² + bx - 84 exactly.

To Find :-

Value of a and b.

Solution :-

The equation  x² + x - 12 divides the equation x³ + ax² + bx - 84 exactly.

Therefore, we have the remainder equals to 0.

On Dividing, the given equation by x² + x - 12, we get  

(a - 8)x² + (b + 5)x = 0

On comparing the equation with the given equation, we get

a = 8 and b = -5

Hence, the values of a and b are 8 and -5.

Answer 2

Ques:-If x² + x - 12 divides P(x) = x + ax² + bx - 84 exactly, find a and b.

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$\large\bold{\underline{\underline{According\:to\:question:}}}$

$x {}^{2} + x - 12 = x {}^{2} + 4x - 3x - 12 \\ \\ = x(x + 4) - 3(x + 4)$

$= (x + 4) \: (x - 3)$

Given p(x) = $x^3+ ax^2 + bx -84$

$x^2+x-12$ divides exactly,

Therefore,

$\implies$$x^2+x-12$ is a factor of p(x)

$\implies$$(x+4)\:(x-3)$ is a factor of p(x)

$\implies$$x+4$ and $x-3$ are both factors of p(x)

$\implies$$p(-4)=0$ and $p(3)=0$

$( - 4) {}^{3} + 9( - 4) {}^{2} + b( - 4) - 84 = 0 \\ \\ 3 {}^{3} + a \times 3 {}^{2} + b \times 3 - 84 = 0 \\ \\ - 64 + 16a - 4b - 84 = 0 \\ \\ 27 + 9a + 3b - 84 = 0 \\ \\ 16a - 4b - 148 = 0 \\ \\ 9a + 3b - 57 = 0 \\ \\ 4a - b - 37 = 0 \\ \\ 3a + b - 19 = 0$

$\large\bold{\underline{On\:adding\:equation\:(1)\:and\:(2)}}$

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$7a-56=0$ $\implies$ $a=8$

From (ii),

$3\times8+b-19=0$ $\implies$ $b= -5$

Hence,

$\implies$$\bold{\red{a=8}}$

$\implies$$\bold{\red{b=-5}}$