Find the zeroes of the polynomial $p(x) = 3x^2 - 4x - 4$. Hence, write a polynomial whose each of the zeroes is 2 more than zeroes of $p(x)$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:36 · grounding rag
Model Answer
Finding zeroes of p(x) = 3x² – 4x – 4:
Splitting the middle term:
$$3x^2 - 4x - 4 = 3x^2 - 6x + 2x - 4 = 3x(x-2) + 2(x-2) = (3x+2)(x-2)$$
Zeroes: $3x + 2 = 0 \Rightarrow x = -\dfrac{2}{3}$, and $x - 2 = 0 \Rightarrow x = 2$
So, $\alpha = -\dfrac{2}{3}$, $\beta = 2$.
New polynomial with zeroes 2 more than above:
New zeroes: $\alpha' = -\dfrac{2}{3} + 2 = \dfrac{4}{3}$ and $\beta' = 2 + 2 = 4$
Sum of new zeroes $= \dfrac{4}{3} + 4 = \dfrac{16}{3}$
Product of new zeroes $= \dfrac{4}{3} \times 4 = \dfrac{16}{3}$
Required polynomial $= x^2 - \dfrac{16}{3}x + \dfrac{16}{3}$ or $\mathbf{3x^2 - 16x + 16}$.
Source: Chapter 2, Section 2.3
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Explanation
- Factorisation by splitting is the standard method for finding zeroes; examiners expect the middle-term split shown clearly.
- For the second part, add 2 to each zero, then use sum and product to form the new polynomial: $k[x^2 - (\text{sum})x + (\text{product})]$. Taking $k = 3$ clears fractions — a neat finish examiners appreciate.
- Write all steps: splitting → factors → zeroes → new sum/product → new polynomial. Skipping steps costs marks.