Question

Two Algebra Questions! Help!

1) Divide 8x4 – 6x3 + 7x2 – 11x +10 by 2x – 1 using long division. Show all work. Then explain if 2x – 1 is a factor of the dividend.

2) Factor f(x) = x4 + x3 – 8x2 + 6x + 36 completely. Show all work for finding the factors. Sketch the graph by hand, use the graph below, label at least 6 points on the graph. State the factors and the roots. (Hint: two of your factors will be complex

Answer 1

ANSWER TO QUESTION 1

Check attachment for long division.

From our long division we can write the following;

`(8x^4-6x^3+7x^2-11x+10)/(2x-1) =4x^3-x^2+3x-4+(6)/(2x-1)`

Since the polynomial

`8x^4-6x^3+7x^2-11x+10` leaves a non zero remainder of
`6` when divided by
`2x-1`, we conclude that
`2x-1` not a factor of the dividend,

ANSWER TO QUESTION 2

We want to factor

`f(x)=x^4+x^3-8x^2+6x+36`

The possible rational roots are;

`\pm1, \pm 2,\pm 3, \pm 4,\pm 6,\pm 9, \pm18, \pm 36`

We found

`f(-2)=(-2)^4+(-2)^3-8(-2)^2+6(-2)+36`

`f(-2)=16+-8-32+-12+36`

`f(-2)=-36+36`

`f(-2)=0`

and

`f(-3)=(-3)^4+(-3)^3-8(-3)^2+6(-3)+36`

`f(-3)=81-27-72-18+36`

`f(-3)=-36+36`

`f(-3)=0`.

This means that
`(x+2)` and
`(x+3)` are factors of the polynomial.

This also means that

`(x+2)(x+3)=x^2+5x+6` is also a factor of the polynomial

So we apply long division to obtain the remaining factors as shown in the attachment.

`\Rightarrow f(x)=x^4+x^3-8x^2+6x+36=(x+2)(x+3)(x^2-4x+6)`

We factor further to obtain;

`\Rightarrow f(x)=x^4+x^3-8x^2+6x+36=(x+2)(x+3)(x-(2-√(2)i))(x-(2+√(2)i))`