Question

If (x-2) and (X+3) are factors of x cube + a x square + bx - 30 find a and b

Answer 1

Answer:

a = 6

b = -1

Step-by-step explanation:

Given:

x - 2 and x + 3 are the factors of x³ + ax² + bx - 30

To find:

The value of a and b

Solution:

Since, x - 2 is a factor of equation.

Therefore, x - 2 = 0 ⟹ x = 2

Now,

$x^3+ ax^2+ bx - 30 \\ \\ {(2)}^{3} + a {(2)}^{2} + b.2 - 30 = 0 \\ \\ 8 + 4.a + 2b = 30 \\ \\ 4a + 2b = 30 - 8 \\ \\ 2(2a + b) = 22 \\ \\ 2a + b = 11 \\ \\ b = 11 - 2a$

Again, x + 3 is factor.

Therefore, x + 3 = 0 ⟹ x = - 3

So,

$x^3+ ax^2+ bx - 30 \\ \\ {( - 3)}^{3} + a( - 3) {}^{2} + b( - 3) - 30 = 0 \\ \\ - 27 + a.9 - 3b = 30 \\ \\ 9a - 3b = 30 + 27 \\ \\ 3(3a - b)= 57\\ \\ [\text{Putting the value of b}] \\ \\ 3a - (11 - 2a) = 19 \\ \\ 3a - 11 + 2a = 19 \\ \\ 5a = 19 + 11 \\ \\ 5a = 30 \\ \\ a = 6 \\ \\ b = 11 - 2a = 11 - 12 \\ \\b \implies - 1$

Hence, The required value of a = 6 and b = - 1 .