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Algebra

Question 3 of 8

Algebraic Division

Question

Use long division to divide \((5x^4 + 12x^3 - 25x^2 + 75)\) by \((x + 3)\)

Show solution Hide solution Fully worked — 5 steps
  1. Find the first term of the quotient

    Divide \(5x^4\) by \(x\) to get the first term of the quotient, \(5x^3\). Multiply the divisor \((x + 3)\) by \(5x^3\) and subtract:

    \[ 5x^4 \div x = 5x^3 \]
    \[ (5x^3)(x + 3) = 5x^4 + 15x^3 \]
    \[ (5x^4 + 12x^3) - (5x^4 + 15x^3) = -3x^3 \]

    Bring down the next term, \(-25x^2\); the working line is now \(-3x^3 - 25x^2\).

  2. Find the second term of the quotient

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

    \[ -3x^3 \div x = -3x^2 \]
    \[ (-3x^2)(x + 3) = -3x^3 - 9x^2 \]
    \[ (-3x^3 - 25x^2) - (-3x^3 - 9x^2) = -16x^2 \]

    Bring down the next term, \(0\); the working line is now \(-16x^2 + 0\).

  3. Find the third term of the quotient

    Divide \(-16x^2\) by \(x\) to get the third term of the quotient, \(-16x\). Multiply the divisor \((x + 3)\) by \(-16x\) and subtract:

    \[ -16x^2 \div x = -16x \]
    \[ (-16x)(x + 3) = -16x^2 - 48x \]
    \[ (-16x^2 + 0) - (-16x^2 - 48x) = 48x \]

    Bring down the next term, \(75\); the working line is now \(48x + 75\).

  4. Find the fourth term of the quotient

    Divide \(48x\) by \(x\) to get the fourth term of the quotient, \(48\). Multiply the divisor \((x + 3)\) by \(48\) and subtract:

    \[ 48x \div x = 48 \]
    \[ (48)(x + 3) = 48x + 144 \]
    \[ (48x + 75) - (48x + 144) = -69 \]
  5. State the result

    In this case the remainder is \(-69\). Therefore:

    \[ (5x^4 + 12x^3 - 25x^2 + 0 + 75) \div (x + 3) = 5x^3 - 3x^2 - 16x + 48 - \frac{69}{x + 3} \]

    That is, the quotient is \(5x^3 - 3x^2 - 16x + 48\) and the remainder is \(-69\).