Question

Use polynomial long division to write each of the following ratios in q(x)+ r/x-a form, where q(x) is a polynomial and r is the remainder. (a) (x^2-6x+11)/x-4 (b) (x^2+2x-25)/x+7 (c) (3x^2+17x+25)/x+4 (d) (5x^2-41x+3)/x-8

Answer 1

To write each of the given ratios in the form q(x)+ r/(x-a), we will use polynomial long division.

Step-by-step explanation:

In order to write each of the given ratios in the form q(x)+ r/(x-a), we will use polynomial long division.

(a) (x^2-6x+11)/(x-4)

1. Divide x^2 by x to get x. Multiply x-4 by x to get x^2-4x.
2. Subtract x^2-4x from x^2-6x+11 to get -2x+11.
3. Divide -2x by x to get -2. Multiply x-4 by -2 to get -2x+8.
4. Subtract -2x+8 from -2x+11 to get 3. Since there are no more terms to divide by x, 3 is the remainder.

Therefore, the ratio (x^2-6x+11)/(x-4) can be written as x-2+3/(x-4).

(b) (x^2+2x-25)/(x+7)

1. Divide x^2 by x to get x. Multiply x+7 by x to get x^2+7x.
2. Subtract x^2+7x from x^2+2x-25 to get -5x-25.
3. Divide -5x by x to get -5. Multiply x+7 by -5 to get -5x-35.
4. Subtract -5x-35 from -5x-25 to get 10. Since there are no more terms to divide by x, 10 is the remainder.

Therefore, the ratio (x^2+2x-25)/(x+7) can be written as x-5+10/(x+7).

(c) (3x^2+17x+25)/(x+4)

1. Divide 3x^2 by x to get 3x. Multiply x+4 by 3x to get 3x^2+12x.
2. Subtract 3x^2+12x from 3x^2+17x+25 to get 5x+25.
3. Divide 5x by x to get 5. Multiply x+4 by 5 to get 5x+20.
4. Subtract 5x+20 from 5x+25 to get 5. Since there are no more terms to divide by x, 5 is the remainder.

Therefore, the ratio (3x^2+17x+25)/(x+4) can be written as 3x+5+5/(x+4).

(d) (5x^2-41x+3)/(x-8)

1. Divide 5x^2 by x to get 5x. Multiply x-8 by 5x to get 5x^2-40x.
2. Subtract 5x^2-40x from 5x^2-41x+3 to get x+3.
3. Divide x by x to get 1. Multiply x-8 by 1 to get x-8.
4. Subtract x-8 from x+3 to get 11. Since there are no more terms to divide by x, 11 is the remainder.

Therefore, the ratio (5x^2-41x+3)/(x-8) can be written as 5x+1+11/(x-8).