5 The equation
$$z ^ { 3 } + 2 z ^ { 2 } - 5 z - 3 = 0$$
has roots \(\alpha , \beta\) and \(\gamma\)
Find a cubic equation with roots
$$\frac { 1 } { 2 } \alpha - 1 , \frac { 1 } { 2 } \beta - 1 \text { and } \frac { 1 } { 2 } \gamma - 1$$
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Question 5:
Answer Marks
Guidance
\(w = \frac{1}{2}z - 1\) so \(z = 2w + 2\) M1
Expresses \(w\) in terms of \(z\), or states values of sum, product and pairwise sum of roots; condone sign errors
Correct value \((-4)\) of sum of roots of new equation A1
Expresses \(z\) correctly in terms of \(w\), or obtains correct value
\((2w+2)^3 + 2(2w+2)^2 - 5(2w+2) - 3 = 0\) M1
Substitutes expression for \(z\) into original equation
\(8w^3 + 24w^2 + 24w + 8 + 8w^2 + 16w + 8 - 10w - 10 - 3 = 0\) M1
Simplifies equation in \(w\)
\(8w^3 + 32w^2 + 30w + 3 = 0\) A1
Correct equation (any correct form)
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## Question 5:
$w = \frac{1}{2}z - 1$ so $z = 2w + 2$ | M1 | Expresses $w$ in terms of $z$, or states values of sum, product and pairwise sum of roots; condone sign errors
Correct value $(-4)$ of sum of roots of new equation | A1 | Expresses $z$ correctly in terms of $w$, or obtains correct value
$(2w+2)^3 + 2(2w+2)^2 - 5(2w+2) - 3 = 0$ | M1 | Substitutes expression for $z$ into original equation
$8w^3 + 24w^2 + 24w + 8 + 8w^2 + 16w + 8 - 10w - 10 - 3 = 0$ | M1 | Simplifies equation in $w$
$8w^3 + 32w^2 + 30w + 3 = 0$ | A1 | Correct equation (any correct form)
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5 The equation
$$z ^ { 3 } + 2 z ^ { 2 } - 5 z - 3 = 0$$
has roots $\alpha , \beta$ and $\gamma$\\
Find a cubic equation with roots
$$\frac { 1 } { 2 } \alpha - 1 , \frac { 1 } { 2 } \beta - 1 \text { and } \frac { 1 } { 2 } \gamma - 1$$
\hfill \mbox{\textit{AQA Further Paper 2 2021 Q5 [5]}}