Standard High contrast

Algebra

Question 3 of 6

Remainder Theorem

Question

The expression, \(x^3 + ax^2 - 2x - 4\), is divisible exactly by \((x + 1)\). Find the value of \(a\) and the remainder when the expression is divided by \((x + 2)\)

Show solution Hide solution Fully worked — 8 steps
  1. Set up the function and the values to substitute

    Let \(f(x) = x^3 + ax^2 - 2x - 4\). Since \((x + 1)\) is a factor of \(x^3 + ax^2 - 2x - 4\), we have \(f(-1) = 0\). The second divisor is \(x + 2\), so the required remainder is \(f(-2)\). We need to find \(a\) and \(f(-2)\).

  2. To find \(a\): substitute \(x = -1\) into \(f(x)\)
    \[ f(-1) = (-1)^3 + a(-1)^2 - 2(-1) - 4 \]
  3. Solve for \(f(-1)\)
    \[ f(-1) = -1 + a + 2 - 4 \]
    \[ f(-1) = a - 3 \]
  4. Equate \(f(-1) = a - 3\) and \(f(-1) = 0\), then solve for \(a\)
    \[ a - 3 = 0 \]
    \[ a = 3 \]
  5. To find \(f(-2)\): substitute \(a = 3\) into \(f(x)\)
    \[ f(x) = x^3 + 3x^2 - 2x - 4 \]
  6. Substitute \(x = -2\) into \(f(x)\)
    \[ f(-2) = (-2)^3 + 3(-2)^2 - 2(-2) - 4 \]
  7. Solve for \(f(-2)\)
    \[ f(-2) = -8 + 12 + 4 - 4 \]
    \[ f(-2) = 4 \]
  8. Final answer

    That is, \(a\) is 3 and the remainder on dividing by \((x + 2)\) is 4.

    \[ a = 3 \quad \text{and} \quad f(-2) = 4 \]