if x^2 +x-12 is a factor of x^3+ax^2-10x-b then find the values of a and b

Question

if x^2 +x-12 is a factor of x^3+ax^2-10x-b then find the values of a and b

in progress 0
Iris 2 weeks 2021-09-12T08:44:48+00:00 1 Answer 0

Answers ( )

    0
    2021-09-12T08:46:12+00:00

    Answer:

    a=3

    b=24

    Step-by-step explanation:

    If x^2+x-12 is a factor of x^3+ax^2-10x-b, then the factors of x^2+x-12 must also be factors of x^3+ax^2-10x-b.

    So what are the factors of x^2+x-12?  Well the cool thing here is the coefficient of x^2[tex] is 1 so all we have to look for are two numbers that multiply to be -12 and add to be positive 1 which in this case is 4 and -3.-12=4(-3) while 1=4+(-3).So the factored form of [tex]x^2+x-12 is (x+4)(x-3).

    The zeros of x^2+x-12 are therefore x=-4 and x=3.  We know those are zeros of x^2+x-12 by the factor theorem.  

    So x=-4 and x=3 are also zeros of x^3+ax^2-10x-b because we were told that x^2+x-12 was a factor of it.

    This means that when we plug in -4, the result will be 0. It also means when we plug in 3, the result will be 0.

    Let’s do that.

    (-4)^3+a(-4)^2-10(-4)-b=0  Equation 1.

    (3)^3+a(3)^2-10(3)-b=0  Equation 2.

    Let’s simplify Equation 1 a little bit:

    (-4)^3+a(-4)^2-10(-4)-b=0

    -64+16a+40-b=0

    -24+16a-b=0

    16a-b=24

    Let’s simplify Equation 2 a little bit:

    (3)^3+a(3)^2-10(3)-b=0

    27+9a-30-b=0

    -3+9a-b=0

    9a-b=3

    So we have a system of equations to solve:

    16a-b=24

    9a-b=3

    ———- This is setup for elimination because the b’s are the same. Let’s subtract the equations.

    16a-b=24

    9a-b=   3

    ——————Subtracting now!

    7a    =21

    Divide both sides by 7:

     a   =3

    Now use one the equations with a=3 to find b.

    How about 9a-b=3 with a=3.

    So plug in 3 for a.

    9a-b=3

    9(3)-b=3

    27-b=3

    Subtract 27 on both sides:

       -b=-24

    Multiply both sides by -1:

        b=24

    So a=3 and b=24

Leave an answer

Browse
Browse

27:3+15-4x7+3-1=? ( )