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Algebra

Question 5 of 8

Algebraic Division

Question

Use long division to divide \((2x^3 - 5x^2 + 15)\) by \((2x + 3)\)

Show solution Hide solution Fully worked — 4 steps
  1. Find the first term of the quotient

    Divide \(2x^3\) by \(2x\) to get the first term of the quotient, \(x^2\). Multiply the divisor \((2x + 3)\) by \(x^2\) and subtract:

    \[ 2x^3 \div 2x = x^2 \]
    \[ (x^2)(2x + 3) = 2x^3 + 3x^2 \]
    \[ (2x^3 - 5x^2) - (2x^3 + 3x^2) = -8x^2 \]

    Bring down the next term, \(0\); the working line is now \(-8x^2 + 0\).

  2. Find the second term of the quotient

    Divide \(-8x^2\) by \(2x\) to get the second term of the quotient, \(-4x\). Multiply the divisor \((2x + 3)\) by \(-4x\) and subtract:

    \[ -8x^2 \div 2x = -4x \]
    \[ (-4x)(2x + 3) = -8x^2 - 12x \]
    \[ (-8x^2 + 0) - (-8x^2 - 12x) = 12x \]

    Bring down the next term, \(15\); the working line is now \(12x + 15\).

  3. Find the third term of the quotient

    Divide \(12x\) by \(2x\) to get the third term of the quotient, \(6\). Multiply the divisor \((2x + 3)\) by \(6\) and subtract:

    \[ 12x \div 2x = 6 \]
    \[ (6)(2x + 3) = 12x + 18 \]
    \[ (12x + 15) - (12x + 18) = -3 \]
  4. State the result

    In this case the remainder is \(-3\). Therefore:

    \[ (2x^3 - 5x^2 + 15) \div (2x + 3) = x^2 - 4x + 6 - \frac{3}{2x + 3} \]

    That is, the quotient is \(x^2 - 4x + 6\) and the remainder is \(-3\).