SSCE HSC Mathematics Extension 2 Polynomial equations: The cubic equation $x^3 + 2x^2 +

The cubic equation $x^3 + 2x^2 + 5x - 1 = 0$ has roots $\alpha, \beta$ and $\gamma$ - HSC - SSCE Mathematics Extension 2 - Question 3 - 2018 - Paper 1

Question 3

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The cubic equation $x^3 + 2x^2 + 5x - 1 = 0$ has roots $\alpha, \beta$ and $\gamma$. Which cubic equation has roots $-\frac{1}{\alpha}, -\frac{1}{\beta}, -\frac{1}{... show full transcript

Worked Solution & Example Answer:The cubic equation $x^3 + 2x^2 + 5x - 1 = 0$ has roots $\alpha, \beta$ and $\gamma$ - HSC - SSCE Mathematics Extension 2 - Question 3 - 2018 - Paper 1

Step 1

Identify the given roots

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Answer

The roots of the original cubic equation are $\alpha, \beta, \gamma$ . We need to find a cubic equation whose roots are the reciprocals of these roots, changed to negatives: $-\frac{1}{\alpha}, -\frac{1}{\beta}, -\frac{1}{\gamma}$ .

Step 2

Use the relationship between roots and coefficients

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Answer

For a cubic equation of the form $x^3 + px^2 + qx + r = 0$ , if the roots are $r_1, r_2, r_3$ , then:

$p = -(r_1 + r_2 + r_3)$
$q = r_1r_2 + r_2r_3 + r_1r_3$
$r = -r_1r_2r_3$ .

For the new roots, we can use these relationships to derive the new coefficients.

Step 3

Determine new coefficients

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Answer

For the roots $-\frac{1}{\alpha}, -\frac{1}{\beta}, -\frac{1}{\gamma}$ , we have:

The sum of the new roots: $-\frac{1}{\alpha} - \frac{1}{\beta} - \frac{1}{\gamma} = -\frac{\beta\gamma + \alpha\gamma + \alpha\beta}{\alpha\beta\gamma}$
The product of the new roots: $-\frac{1}{\alpha} \cdot -\frac{1}{\beta} \cdot -\frac{1}{\gamma} = \frac{1}{\alpha\beta\gamma}$

From the original polynomial, we have $\alpha + \beta + \gamma = -2$ , $\alpha\beta + \beta\gamma + \gamma\alpha = 5$ , and $\alpha\beta\gamma = 1$ .

Step 4

Construct the new cubic equation

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Answer

By substituting the calculated sums and products into the polynomial form:

The coefficient of $x^2$ (new $p$ ) is: $\frac{5}{1} = 5$
The coefficient of $x$ (new $q$ ) is: $-(-2) = 2$
The constant term (new $r$ ) is: $-\frac{1}{1} = -1$

Thus, the new cubic equation is: $x^3 + 5x^2 + 2x - 1 = 0$

Step 5

Select the correct option

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Answer

Comparing with the provided options, the correct cubic equation is option D: $x^3 + 5x^2 - 2x + 1 = 0$ .

The cubic equation $x^3 + 2x^2 + 5x - 1 = 0$ has roots $\alpha, \beta$ and $\gamma$ - HSC - SSCE Mathematics Extension 2 - Question 3 - 2018 - Paper 1

Worked Solution & Example Answer:The cubic equation $x^3 + 2x^2 + 5x - 1 = 0$ has roots $\alpha, \beta$ and $\gamma$ - HSC - SSCE Mathematics Extension 2 - Question 3 - 2018 - Paper 1

Identify the given roots

Use the relationship between roots and coefficients

Determine new coefficients

Construct the new cubic equation

Select the correct option

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