Answer
423.9k+ views
Hint: Zeroes of the polynomial are the real values of the variable for which the value of polynomial becomes zero. You can factorize the given polynomial and put the factors equal to zero. Then find the values of the variable x.
Complete step by step answer:
Given polynomial is-${{\text{x}}^2} - {\text{x}} - 6$. Now let f(x) =${{\text{x}}^2} - {\text{x}} - 6$
We have to find the zeroes of the given polynomial. Zeroes of the polynomial are the real values of the variable for which the value of polynomial becomes zero. So to find the zeroes of f(x), put f(x)$ = 0$.
$ \Rightarrow {{\text{x}}^2} - {\text{x}} - 6 = 0$
Now we will factorize the polynomial by splitting the middle term as we can write $ - {\text{x = - 3x + 2x}}$, so we get-
$ \Rightarrow {{\text{x}}^2} - 3{\text{x + 2x - 6 = 0}} \Rightarrow \left( {{{\text{x}}^2} - 3{\text{x}}} \right){\text{ + }}\left( {{\text{2x - 6}}} \right){\text{ = 0}}$
On taking x and $2$ common, we get-
$ \Rightarrow {\text{x}}\left( {{\text{x}} - 3} \right) + 2\left( {{\text{x - 3}}} \right) = 0$
By separating the common factor we get,
$
\Rightarrow \left( {{\text{x - 3}}} \right) \times \left( {{\text{x + 2}}} \right) = 0 \\
\Rightarrow \left( {{\text{x - 3}}} \right) = 0{\text{ or }}\left( {{\text{x + 2}}} \right) = 0 \\
$
Now equating the values of x to zero, find the value of x,
$ \Rightarrow {\text{x}} = 3{\text{ or x = }} - 2$
Therefore, the zeroes of the polynomial ${{\text{x}}^2} - {\text{x}} - 6$ are $ - 2,3$ .
Hence, the answer is ‘C’.
Note: You can also use the formula ${\text{x = }}\dfrac{{ - {\text{b}} \pm \sqrt {{{\text{b}}^2} - 4{\text{ac}}} }}{{2{\text{a}}}}$ to find the zeroes of quadratic polynomial ${\text{a}}{{\text{x}}^2}{\text{ + bx + c}}$. This formula is known as discriminant formula. You just need to put the values and simplify. Here, b=$ - 1$, a=$1$ and c=$ - 6$. On putting these values in formula, we get-
$ \Rightarrow {\text{x}} = \dfrac{{ - \left( { - 1} \right) \pm \sqrt {{{\left( { - 1} \right)}^2} - 4 \times 1 \times \left( { - 6} \right)} }}{{2 \times 1}} = \dfrac{{1 \pm \sqrt {1 + 24} }}{2}$
On simplifying we get-
\[
\Rightarrow {\text{x = }}\dfrac{{1 \pm \sqrt {25} }}{2} = \dfrac{{1 \pm 5}}{2} \\
\Rightarrow {\text{x = }}\dfrac{{1 + 5}}{2}{\text{ or x = }}\dfrac{{1 - 5}}{2} \\
\Rightarrow {\text{x = -2 or 3}} \\
\]
Hence we get the same answer.
Complete step by step answer:
Given polynomial is-${{\text{x}}^2} - {\text{x}} - 6$. Now let f(x) =${{\text{x}}^2} - {\text{x}} - 6$
We have to find the zeroes of the given polynomial. Zeroes of the polynomial are the real values of the variable for which the value of polynomial becomes zero. So to find the zeroes of f(x), put f(x)$ = 0$.
$ \Rightarrow {{\text{x}}^2} - {\text{x}} - 6 = 0$
Now we will factorize the polynomial by splitting the middle term as we can write $ - {\text{x = - 3x + 2x}}$, so we get-
$ \Rightarrow {{\text{x}}^2} - 3{\text{x + 2x - 6 = 0}} \Rightarrow \left( {{{\text{x}}^2} - 3{\text{x}}} \right){\text{ + }}\left( {{\text{2x - 6}}} \right){\text{ = 0}}$
On taking x and $2$ common, we get-
$ \Rightarrow {\text{x}}\left( {{\text{x}} - 3} \right) + 2\left( {{\text{x - 3}}} \right) = 0$
By separating the common factor we get,
$
\Rightarrow \left( {{\text{x - 3}}} \right) \times \left( {{\text{x + 2}}} \right) = 0 \\
\Rightarrow \left( {{\text{x - 3}}} \right) = 0{\text{ or }}\left( {{\text{x + 2}}} \right) = 0 \\
$
Now equating the values of x to zero, find the value of x,
$ \Rightarrow {\text{x}} = 3{\text{ or x = }} - 2$
Therefore, the zeroes of the polynomial ${{\text{x}}^2} - {\text{x}} - 6$ are $ - 2,3$ .
Hence, the answer is ‘C’.
Note: You can also use the formula ${\text{x = }}\dfrac{{ - {\text{b}} \pm \sqrt {{{\text{b}}^2} - 4{\text{ac}}} }}{{2{\text{a}}}}$ to find the zeroes of quadratic polynomial ${\text{a}}{{\text{x}}^2}{\text{ + bx + c}}$. This formula is known as discriminant formula. You just need to put the values and simplify. Here, b=$ - 1$, a=$1$ and c=$ - 6$. On putting these values in formula, we get-
$ \Rightarrow {\text{x}} = \dfrac{{ - \left( { - 1} \right) \pm \sqrt {{{\left( { - 1} \right)}^2} - 4 \times 1 \times \left( { - 6} \right)} }}{{2 \times 1}} = \dfrac{{1 \pm \sqrt {1 + 24} }}{2}$
On simplifying we get-
\[
\Rightarrow {\text{x = }}\dfrac{{1 \pm \sqrt {25} }}{2} = \dfrac{{1 \pm 5}}{2} \\
\Rightarrow {\text{x = }}\dfrac{{1 + 5}}{2}{\text{ or x = }}\dfrac{{1 - 5}}{2} \\
\Rightarrow {\text{x = -2 or 3}} \\
\]
Hence we get the same answer.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)