Polynomials: Beyond the Parabola (~15 marks, Paper 1)

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In Grade 12, we learn to handle functions with powers higher than 2 (like $x^3$). Polynomial questions appear in Paper 1 under Algebra (~25 marks) and are also essential for the Calculus section (sketching cubic graphs).

The Key Idea: Factors = Roots = $x$-intercepts

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Finding the factors of a polynomial is the same as finding the $x$-intercepts of the graph:

• If $(x - 2)$ is a factor of $f(x)$, then $f(2) = 0$, and the graph crosses the $x$-axis at $x = 2$.
• This connection is what makes the Factor Theorem so powerful.

The Two Theorems

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How to Use the Factor Theorem

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To factorise $f(x) = x^3 - 7x - 6$:

1. Try values: Test $f(1)$, $f(-1)$, $f(2)$, $f(-2)$, $f(3)$, $f(-3)$, $f(6)$, $f(-6)$…
2. Find a root: $f(-1) = -1 + 7 - 6 = 0$ ✓ → so $(x + 1)$ is a factor
3. Divide: Use long division or synthetic division to get the quadratic quotient
4. Factorise the quadratic: $x^3 - 7x - 6 = (x + 1)(x^2 - x - 6) = (x + 1)(x - 3)(x + 2)$

💡 Which values to try: Always try the factors of the constant term ($\pm 1, \pm 2, \pm 3, \pm 6$ for $-6$). One of these will be a root.

Synthetic Division (The Quick Method)

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For dividing $f(x)$ by $(x - a)$:

1. Write down the coefficients of $f(x)$
2. Bring down the first coefficient
3. Multiply by $a$, add to the next coefficient, repeat
4. The last number is the remainder

This is faster than long division and less error-prone.

Deep Dives (click into each)

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• Remainder & Factor Theorems — the tools for testing factors and predicting remainders
• Solving Cubic Equations — the complete strategy from finding the first root to full factorisation

🚨 Common Mistakes

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1. Not trying enough values: If $f(1) \neq 0$, try $f(-1)$, $f(2)$, etc. Be systematic — try all factors of the constant term.
2. Division errors: In long division, make sure you account for “missing” terms. If $f(x) = x^3 + 2x - 5$, write it as $x^3 + 0x^2 + 2x - 5$ (include the $0x^2$ placeholder).
3. Forgetting to factorise the quadratic: After finding the first factor and dividing, you get a quadratic — you must still factorise (or use the formula on) that quadratic.
4. Sign errors in the Factor Theorem: $(x + 1)$ is a factor means $f(-1) = 0$, NOT $f(1) = 0$. The sign flips!
5. Not linking to graphs: If a question asks for the $x$-intercepts of a cubic, you’re really being asked to factorise and solve $f(x) = 0$.

🔗 Related topics:

• Differential Calculus — factorising cubics is essential for finding $x$-intercepts when sketching cubic graphs
• Algebra — cubic equations appear in Question 1 of Paper 1

📌 Grade 10 foundation: Factorisation — the factoring toolkit you must know before tackling cubics

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