Questions: Use long division to divide. (5x^3-15-20x+x^4) ÷ (x^2-x-3)

Transcript text: Use long division to divide. \[ \left(5 x^{3}-15-20 x+x^{4}\right) \div\left(x^{2}-x-3\right) \]

Solution

Solution Steps

Step 1: Arrange the Polynomials in Descending Order

First, arrange the dividend and divisor in descending order of powers: \[ \text{Dividend: } x^4 + 5x^3 - 20x - 15 \] \[ \text{Divisor: } x^2 - x - 3 \]

Step 2: Perform Polynomial Division

To perform the division, we divide the leading term of the dividend by the leading term of the divisor and then multiply the entire divisor by this result. We subtract this from the dividend to get a new dividend and repeat the process.

Divide \(x^4\) by \(x^2\): \[ \frac{x^4}{x^2} = x^2 \] Multiply the divisor by \(x^2\): \[ x^2(x^2 - x - 3) = x^4 - x^3 - 3x^2 \] Subtract from the dividend: \[ (x^4 + 5x^3 - 20x - 15) - (x^4 - x^3 - 3x^2) = 6x^3 + 3x^2 - 20x - 15 \]
Divide \(6x^3\) by \(x^2\): \[ \frac{6x^3}{x^2} = 6x \] Multiply the divisor by \(6x\): \[ 6x(x^2 - x - 3) = 6x^3 - 6x^2 - 18x \] Subtract from the new dividend: \[ (6x^3 + 3x^2 - 20x - 15) - (6x^3 - 6x^2 - 18x) = 9x^2 - 2x - 15 \]
Divide \(9x^2\) by \(x^2\): \[ \frac{9x^2}{x^2} = 9 \] Multiply the divisor by \(9\): \[ 9(x^2 - x - 3) = 9x^2 - 9x - 27 \] Subtract from the new dividend: \[ (9x^2 - 2x - 15) - (9x^2 - 9x - 27) = 7x + 12 \]

Step 3: Identify the Quotient and Remainder

The quotient is: \[ x^2 + 6x + 9 \] The remainder is: \[ 7x + 12 \]