Question

Find the zeros of the polynomial $P(x) = 6{x^2} - 3 - 7x$ .
A) $\dfrac{3}{2}, - \dfrac{1}{3}$
B) $ - \dfrac{3}{2}, - \dfrac{1}{3}$
C) $\dfrac{3}{2},\dfrac{1}{3}$
D) $ - \dfrac{3}{2},\dfrac{1}{3}$

Answer 1

Hint:The zeros of the polynomials indicate the values of the variable that satisfy the given equation when equated to zero. In other words, we have to find the roots of the given polynomial. First write the equation in the standard form and then use the direct formula to obtain the roots.

Complete step-by-step answer:
The given polynomial is $P(x) = 6{x^2} - 3 - 7x$ .
To find the zeros of the above polynomial we will find the roots of the equation $P(x) = 0$ .
Equating the given polynomial to $0$ we write,
$6{x^2} - 3 - 7x = 0$
Observe that the greatest degree of any variable is $2$ that means the given polynomial is a quadratic polynomial and thus the equation is a quadratic equation.
The standard form of the quadratic equation is $a{x^2} + bx + c = 0$ where $a \ne 0$ .
Thus, express the given equation in its standard form as follows:
$6{x^2} - 7x - 3 = 0$
Comparing with the standard form we get $a = 6,b = - 7$ and $c = - 3$ .
The root of the quadratic equation in the general form is given by,
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ … (1)
Using the above obtained values in the equation (1) we get,
$x = \dfrac{{ - \left( { - 7} \right) \pm \sqrt {{{\left( { - 7} \right)}^2} - 4\left( 6 \right)\left( { - 3} \right)} }}{{2\left( 6 \right)}}$
Simplify it further as follows:
$x = \dfrac{{7 \pm \sqrt {49 + 72} }}{{12}}$
Taking the square root and obtaining the roots we get,
$x = \dfrac{{7 \pm 11}}{{12}}$
Thus, the roots of the given equation and thus the zeros of the given polynomial are $x = \dfrac{3}{2}$ and $x = - \dfrac{1}{3}$ .

So, the correct answer is “Option A”.

Note:The given question asks for the zeros of the polynomial, first, one needs to understand that it means that we have to find the roots of the equation that is represented by the polynomial. The roots of the equation depend on the nature of the equation so it is also important to identify the nature of the obtained equation.