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

a=3

b=24

Step-by-step explanation:

If is a factor of , then the factors of must also be factors of .

So what are the factors of ?  Well the cool thing here is the coefficient of is .

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

So x=-4 and x=3 are also zeros of because we were told that 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.

 Equation 1.

 Equation 2.

Let's simplify Equation 1 a little bit:

Let's simplify Equation 2 a little bit:

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

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step-by-step explanation:

step-by-step explanation:

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