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

Question

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

in progress 0
Serenity 3 weeks 2021-10-05T06:53:06+00:00 1 Answer 0 views 0

Answers ( )

    0
    2021-10-05T06:54:19+00:00

    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 .

Leave an answer

Browse
Browse

18:9+8+9*3-7:3-1*13 = ? ( )