Polynomials (Cubic Functions) Grade 12
Mathematics Grade 12 · Term 2 · Weeks 1-2 · 2 questions
Polynomials (Cubic Functions) Questions and Answers Grade 12
2 exam questions on this topic from Mathematics Grade 12 past papers, each with a full answer.
What the ATP covers
Factor and remainder theorems, factorising third-degree polynomials and solving cubic equations.
Textbook: Everything Maths Grade 12, Chapter 5: Polynomials.
Exam Questions
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\[f(1) = 1 - 6 + 9 - 4 = 0\]
Full answer and marking tips
So \((x - 1)\) is a factor. Dividing:
\[f(x) = (x - 4)(x^2 - 2x + 1) = (x - 4)(x - 1)^2\]Comparing with \((x - 4)(x - k)^2\), \(k = 1\)
Answer
\(E(-1\,;\,0)\) is a turning point on the \(x\)-axis, so \((x + 1)\) is a double factor.
\[f(x) = a(x + 1)^2(x - 5)\]Substitute \((0\,;\,-5)\):
\[-5 = a(0 + 1)^2(0 - 5)\]\[-5 = -5a \text{, so } a = 1\]\[f(x) = (x^2 + 2x + 1)(x - 5)\]\[f(x) = x^3 - 3x^2 - 9x - 5\]\(a = 1\), \(b = -3\), \(c = -9\)