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Divide as indicated. Check the answer by showing that the product of the divisor and the​ quotient, plus the​ remainder, is the dividend.

StartFraction 7 x squared minus 23 x minus 20 Over x minus 4 EndFraction

1 answer

To divide \( \frac{7x^2 - 23x - 20}{x - 4} \), we can use polynomial long division.

  1. Long Division Steps:

    • Divide the leading term of the dividend \( 7x^2 \) by the leading term of the divisor \( x \): \[ 7x^2 \div x = 7x \]
    • Multiply the entire divisor \( x - 4 \) by \( 7x \): \[ 7x(x - 4) = 7x^2 - 28x \]
    • Subtract \( 7x^2 - 28x \) from the dividend \( 7x^2 - 23x - 20 \): \[ (7x^2 - 23x - 20) - (7x^2 - 28x) = -23x + 28x - 20 = 5x - 20 \]

  2. Next Step:

    • Now take \( 5x - 20 \) and divide \( 5x \) by \( x \): \[ 5x \div x = 5 \]
    • Multiply \( x - 4 \) by \( 5 \): \[ 5(x - 4) = 5x - 20 \]
    • Subtract: \[ (5x - 20) - (5x - 20) = 0 \]

  3. Result:

    • The quotient is \( 7x + 5 \) and the remainder is \( 0 \).
    • So we have: \[ \frac{7x^2 - 23x - 20}{x - 4} = 7x + 5 \quad \text{with a remainder of } 0. \]

  4. Check the Answer:

    • To verify, we use the formula \( \text{(divisor)} \times \text{(quotient)} + \text{(remainder)} = \text{(dividend)} \): \[ (x - 4)(7x + 5) + 0 \]
      • Multiply it out: \[ = 7x^2 + 5x - 28x - 20 = 7x^2 - 23x - 20 \]
    • This equals the original dividend \( 7x^2 - 23x - 20 \).

Thus, the final answer is: \[ \frac{7x^2 - 23x - 20}{x - 4} = 7x + 5 \] with a remainder of \( 0 \).