Use synthetic division to find the result when 4, x, to the power 4 , minus, 7, x, cubed, plus, 7, x, squared, minus, 12, x, plus, 154x 4 −7x 3 +7x 2 −12x+15 is divided by x, minus, 1x−1. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ b(x)r(x) .
Question
Use synthetic division to find the result when 4, x, to the power 4 , minus, 7, x, cubed, plus, 7, x, squared, minus, 12, x, plus, 154x 4 −7x 3 +7x 2 −12x+15 is divided by x, minus, 1x−1. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ b(x)r(x) .
Solution
To perform synthetic division, we first set up our synthetic division table. We write the coefficients of the polynomial we are dividing (4x^4 - 7x^3 + 7x^2 - 12x + 15) in the top row of the table. We also write the value of x that makes the divisor (x - 1) equal to zero on the left side of the table. In this case, that value is 1.
Our table now looks like this:
1 | 4 -7 7 -12 15
Next, we bring down the first coefficient (4) unchanged:
1 | 4 -7 7 -12 15 |---------------- | 4
Then, we multiply the value on the left (1) by the value we just wrote down (4), and write the result (4) under the second coefficient (-7):
1 | 4 -7 7 -12 15
| 4 |
|---|
| 4 -3 |
We repeat this process for the remaining coefficients. Multiply the value on the left (1) by the value we just wrote down (-3), and write the result (-3) under the next coefficient (7):
1 | 4 -7 7 -12 15
| 4 -3 |
|---|
| 4 -3 4 |
Continue this process:
1 | 4 -7 7 -12 15
| 4 -3 4 -8 |
|---|
| 4 -3 4 -8 7 |
The last number we wrote down (7) is the remainder. The other numbers on the bottom row are the coefficients of the quotient. So, the quotient is 4x^3 - 3x^2 + 4x - 8, and the remainder is 7.
Therefore, the result of the division is 4x^3 - 3x^2 + 4x - 8 + 7/(x - 1).
Similar Questions
Use the long division method to find the result when 4, x, cubed, minus, 4, x, squared, plus, 13, x, minus, 64x 3 −4x 2 +13x−6 is divided by 2, x, minus, 12x−1. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ b(x)r(x) .
Use the long division method to find the result when 8, x, cubed, minus, 2, x, squared, minus, 20, x, plus, 18x 3 −2x 2 −20x+1 is divided by 4, x, minus, 14x−1. If there is a remainder, express the result in the form q, left bracket, x, right bracket, plus, start fraction, r, left bracket, x, right bracket, divided by, b, left bracket, x, right bracket, end fractionq(x)+ b(x)r(x) .
Use the long division method to find the result when 2, x, cubed, minus, 8, x, squared, plus, x, plus, 12x 3 −8x 2 +x+1 is divided by x, minus, 4x−4.
Use the long division method to find the result when 3, x, cubed, plus, 5, x, squared, plus, 3, x, plus, 13x 3 +5x 2 +3x+1 is divided by x, plus, 1x+1.
Use synthetic division to determine the remainder when 5x4 + 7x3 + x2 − 18 is divided by x + 2.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.