Standard High contrast

Algebra

Question 2 of 6

Remainder Theorem

Question

The function \(f(x) = ax^2 + x - 7\) has a remainder of 3 when divided by \((x - 2)\). Find the value of \(a\)

Show solution Hide solution Fully worked — 6 steps
  1. Set up the function and the value to substitute

    Let \(f(x) = ax^2 + x - 7\). The divisor is \(x - 2\), so we evaluate at \(x = 2\). The remainder is 3, so by the Remainder Theorem \(f(2) = 3\). We need to find \(a\).

  2. Substitute \(x = 2\) into \(f(x)\)
    \[ f(2) = a(2)^2 + (2) - 7 \]
  3. Solve for \(f(2)\)
    \[ f(2) = 4a - 5 \]
  4. Equate \(f(2) = 3\) and \(f(2) = 4a - 5\)
    \[ 4a - 5 = 3 \]
  5. Solve for \(a\)
    \[ 4a = 8 \]

    Divide both sides by 4:

    \[ a = 2 \]
  6. Final answer

    That is, \(a\) is 2, and so the function is:

    \[ a = 2 \; \therefore \; f(x) = 2x^2 + x - 7 \]