When is divided by , the remainder is . this result may be expressed as the identity
≡
where q(x) is the quotient.
a) state the degree of q(x).
b) by substituting suitable values of x, find a and b.

a=5

b=-14

I'm not sure if this is what ur lookin for but this is supposed to be the answer

a) Q has degree 1.

b) a=5  while b=-14 ( I did this doing the way the problem suggested and I also did the problem using long division.)

Step-by-step explanation:

a) Q has degree 1.

The left hand side's leading term is .

If we expand (x-1)(x+2) we should see it's leading term is

Since and 2x has degree 1, then our quotient must have degree 1.

Notes: I multiplied (x-1)(x+2) out in part b if you want to see what it looks like totally expanded.

b. a=5  while b=-14

You are given the following identity:

We want to be careful to choose values for x so that the expression for Q(x) doesn't matter.

If x=1, then would be 0 since x-1 is zero at x=1.  

That is plugging in 1 into term gives you:

See what I mean the expression for Q doesn't matter because this result is 0 anyhow.

So let's plug in 1 into both sides:

Now notice the factor x+2 in .

x+2 is 0 when x=-2 since -2+2=0.

So we are going to plug in -2 into both sides:

So the system to solve is:

  a+b=-9

-2a+b=-24

--------------------Subtract the equations to eliminate b:

3a+0=15

3a    =15

Divide both sides by 3:

3a/3=15/3

Simplify both sides:

1a=5

a=5

Using one of the equations we found along with a=5 we can not find b.

a+b=-9 with a=5

5+b=-9

Subtract 5 on both sides:

   b=-9-5

   b=-14

So a=5 while b=-14.

So ax+b is 5x-14.

We could do this another way not suggested by your problem but through long division:

First let's multiply (x-1)(x+2) out using foil.

First: x(x)=x^2

Outer: x(2)=2x

Inner: -1(x)=-x

Last:  -1(2)=-2

-------------------Combine like terms:

x^2+x-2

Now let's do the division:

               2x-6

            ----------------------------

x^2+x-2| 2x^3-4x^2-5x-2

            -(2x^3+2x^2-4x)

              -----------------------

                       -6x^2 - x   -2

                     -(-6x^2-6x+12)

                      --------------------

                                 5x-14

We see the remainder is 5x-14.  This is what we also got doing as your problem suggested using values for x to plug in to find a and b.

Please let me know if something doesn't make sense to you. Have a good day.

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