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The polynomial $x^3 + 2x^2 - 5x - 6$ has zeros $-1$, $-3$ and $\alpha$ - HSC - SSCE Mathematics Extension 1 - Question 1 - 2024 - Paper 1

Question 1

$The-polynomial-$x^3-+-2x^2---5x---6$-has-zeros-$-1$,-$-3$-and-$\alpha$-HSC-SSCE Mathematics Extension 1-Question 1-2024-Paper 1.png$

The polynomial $x^3 + 2x^2 - 5x - 6$ has zeros $-1$, $-3$ and $\alpha$. What is the value of $\alpha$? A. -2 B. 2 C. 3 D. 6

Worked Solution & Example Answer:The polynomial $x^3 + 2x^2 - 5x - 6$ has zeros $-1$, $-3$ and $\alpha$ - HSC - SSCE Mathematics Extension 1 - Question 1 - 2024 - Paper 1

Step 1

Calculate the product of the zeros

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Answer

Since the polynomial is given as $x^3 + 2x^2 - 5x - 6$ , by Vieta's formulas, the product of the zeros (roots) for a cubic polynomial $ax^3 + bx^2 + cx + d$ is given by: $r_1 \cdot r_2 \cdot r_3 = -\frac{d}{a}$ Here, $r_1 = -1$ , $r_2 = -3$ , and $r_3 = \alpha$ . So, $(-1) \cdot (-3) \cdot \alpha = -\frac{-6}{1} = 6$ This simplifies to: $3\alpha = 6$

Step 2

Solve for $\alpha$

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Answer

To find $\alpha$ , we divide both sides by 3: $\alpha = \frac{6}{3} = 2$

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