Math  /  Algebra

Questionx3+2x2+15x5x+3\frac{x^{3}+2 x^{2}+15 x-5}{x+3}

Studdy Solution

STEP 1

1. We are asked to perform polynomial division.
2. The division will be done using synthetic or long division.
3. The divisor is x+3x + 3.

STEP 2

1. Set up the division problem using polynomial long division.
2. Perform the division step-by-step.
3. Simplify the result to find the quotient and remainder.

STEP 3

Set up the polynomial long division. Write the dividend x3+2x2+15x5x^{3} + 2x^{2} + 15x - 5 under the division symbol and the divisor x+3x + 3 outside.
x+3x3+2x2+15x5 \begin{array}{c|c} x + 3 & x^{3} + 2x^{2} + 15x - 5 \\ \end{array}

STEP 4

Divide the first term of the dividend x3x^{3} by the first term of the divisor xx, which gives x2x^{2}. Write x2x^{2} above the division line.
x+3x3+2x2+15x5x2 \begin{array}{c|c} x + 3 & x^{3} + 2x^{2} + 15x - 5 \\ & x^{2} \\ \end{array}

STEP 5

Multiply x2x^{2} by the entire divisor x+3x + 3 and write the result under the dividend, then subtract.
x+3x3+2x2+15x5x2x3+3x2 \begin{array}{c|c} x + 3 & x^{3} + 2x^{2} + 15x - 5 \\ & x^{2} \\ \hline & x^{3} + 3x^{2} \\ \end{array}
Subtract:
(x3+2x2+15x5)(x3+3x2)=x2+15x5 (x^{3} + 2x^{2} + 15x - 5) - (x^{3} + 3x^{2}) = -x^{2} + 15x - 5

STEP 6

Bring down the next term from the dividend, which is 15x15x, and repeat the division process. Divide x2-x^{2} by xx to get x-x.
x+3x3+2x2+15x5x2xx3+3x2x2+15x \begin{array}{c|c} x + 3 & x^{3} + 2x^{2} + 15x - 5 \\ & x^{2} - x \\ \hline & x^{3} + 3x^{2} \\ & -x^{2} + 15x \\ \end{array}

STEP 7

Multiply x-x by the divisor x+3x + 3 and subtract again.
x+3x3+2x2+15x5x2xx3+3x2x2+15xx23x \begin{array}{c|c} x + 3 & x^{3} + 2x^{2} + 15x - 5 \\ & x^{2} - x \\ \hline & x^{3} + 3x^{2} \\ & -x^{2} + 15x \\ & -x^{2} - 3x \\ \end{array}
Subtract:
(x2+15x)(x23x)=18x (-x^{2} + 15x) - (-x^{2} - 3x) = 18x

STEP 8

Bring down the last term, 5-5, and continue the division. Divide 18x18x by xx to get 1818.
x+3x3+2x2+15x5x2x+18x3+3x2x2+15xx23x18x \begin{array}{c|c} x + 3 & x^{3} + 2x^{2} + 15x - 5 \\ & x^{2} - x + 18 \\ \hline & x^{3} + 3x^{2} \\ & -x^{2} + 15x \\ & -x^{2} - 3x \\ & 18x \\ \end{array}

STEP 9

Multiply 1818 by the divisor x+3x + 3 and subtract to find the remainder.
x+3x3+2x2+15x5x2x+18x3+3x2x2+15xx23x18x+54 \begin{array}{c|c} x + 3 & x^{3} + 2x^{2} + 15x - 5 \\ & x^{2} - x + 18 \\ \hline & x^{3} + 3x^{2} \\ & -x^{2} + 15x \\ & -x^{2} - 3x \\ & 18x + 54 \\ \end{array}
Subtract:
(18x5)(18x+54)=59 (18x - 5) - (18x + 54) = -59

STEP 10

The quotient is x2x+18x^{2} - x + 18 and the remainder is 59-59.
The division can be expressed as:
x3+2x2+15x5x+3=x2x+1859x+3 \frac{x^{3} + 2x^{2} + 15x - 5}{x + 3} = x^{2} - x + 18 - \frac{59}{x + 3}

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