MCQ
Choose the correct answer from the given four options in the following questions:A quadratic polynomial, whose zeroes are –3 and 4, is:
- A$x^2-x+12$.
- B$x^2+x+12$.
- ✓$\frac{x^2}{2}-\frac{x}{2}-6$
- D$2 x^2+2 x-24$.
✓
Answer
Correct option: C.
$\frac{x^2}{2}-\frac{x}{2}-6$
Then, sum of zeroes = -3 + 4 = 1
$\Big[\because\ \text{sum of zeroes}=\frac{\text{-b}}{\text{a}}\Big]$
$\Rightarrow\ \frac{-\text{b}}{\text{a}}=\frac{1}{1}\Rightarrow\ \frac{-\text{b}}{\text{c}}=-\frac{(-1)}{1}\ .....(\text{i})$
and product of zeroes = -3 × 4 = -12
$\Big[\because\ \text{product of zeroes} = \frac{\text{c}}{\text{a}}\Big]$
$\Rightarrow\ \frac{\text{c}}{\text{a}}=\frac{-12}{1}\ .....\text{(ii)}$
From Eqs. (i) and (ii),
a = 1, b = -1 and c = -12
$=a x^2+b x+c$
$\therefore \text { Required polynimial }=1 \cdot x^2-1 \cdot x-12$
$=x^2-x-12$
$=\frac{\text{x}^2}{2}-\frac{\text{x}}{2}-6$
We know that, if we multiply/divide and polynomial by any constant, then the zeroes of polynomial do not change.
Alternate Answer
Let the zeroes of a qaudratic polynomial are $\alpha = -3$ and $\beta = 4.$
Then, sum of zeroes $=\alpha + \beta = -3 + 4 = 1$ and product of zeroes $=\alpha\beta = (-3)(4) = -12.$
$\Big[\because\ \text{sum of zeroes}=\frac{\text{-b}}{\text{a}}\Big]$
$\Rightarrow\ \frac{-\text{b}}{\text{a}}=\frac{1}{1}\Rightarrow\ \frac{-\text{b}}{\text{c}}=-\frac{(-1)}{1}\ .....(\text{i})$
and product of zeroes = -3 × 4 = -12
$\Big[\because\ \text{product of zeroes} = \frac{\text{c}}{\text{a}}\Big]$
$\Rightarrow\ \frac{\text{c}}{\text{a}}=\frac{-12}{1}\ .....\text{(ii)}$
From Eqs. (i) and (ii),
a = 1, b = -1 and c = -12
$=a x^2+b x+c$
$\therefore \text { Required polynimial }=1 \cdot x^2-1 \cdot x-12$
$=x^2-x-12$
$=\frac{\text{x}^2}{2}-\frac{\text{x}}{2}-6$
We know that, if we multiply/divide and polynomial by any constant, then the zeroes of polynomial do not change.
Alternate Answer
Let the zeroes of a qaudratic polynomial are $\alpha = -3$ and $\beta = 4.$
Then, sum of zeroes $=\alpha + \beta = -3 + 4 = 1$ and product of zeroes $=\alpha\beta = (-3)(4) = -12.$
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