Question

The binomial $x - 3$ is not a factor of which of the following trinomials?
A. $2{x^2} - x - 15$
B. $3{x^2} - 13x + 12$
C. $2{x^2} - 8x + 6$
D. $2{x^2} - 7x - 3$

Answer 1

Hint: In this question, we are given a monomial factor of the given trinomials, out of which one is incorrect i.e., $x - 3$ is not a factor one of the given trinomials. So, we have to find that trinomial.
The property which we will use in this question is as follows: If $x = a$ is a factor of any polynomial $p(x)$ , then, we have $p(a) = 0$ , i.e., if we put $a$ in place $x$ , then, it will give zero.
So, we will put $x = 3$ one- by- one in each trinomial and check which trinomial will not give zero.

Complete step-by-step answer:
We are given a monomial $x - 3$ and four trinomials.
To check out $x - 3$ is not a factor of the given trinomials.
We know that, if $x = a$ is a factor of any polynomial $p(x)$ , then, we have $p(a) = 0$ .
So, we have the factor $x - 3$ , we will first put it equal to zero and find the value of $x$ , i.e., $x - 3 = 0$ gives $x = 3$ .
Now, we will put this value of $x$ , one- by- one in each of the given trinomials.

First trinomial is $2{x^2} - x - 15$ , putting $x = 3$ , we get, $2{(3)^2} - 3 - 15$ . On solving, we get $18 - 3 - 15 = 18 - 18 = 0$ .
Hence, $x - 3$ is a factor of the trinomial $2{x^2} - x - 15$ .

Now, the next trinomial is $3{x^2} - 13x + 12$ , putting $x = 3$ , we get, $3{(3)^2} - 13(3) + 12$ . On solving, we get $27 - 39 + 12 = 39 - 39 = 0$ .
Hence, $x - 3$ is a factor of the trinomial $3{x^2} - 13x + 12$ .

Now, the next trinomial is $2{x^2} - 8x + 6$ , putting $x = 3$ , we get, $2{(3)^2} - 8(3) + 6$ . On solving, we get $18 - 24 + 6 = 24 - 24 = 0$ .
Hence, $x - 3$ is a factor of the trinomial $2{x^2} - 8x + 6$ .

Finally, we have the trinomial $2{x^2} - 7x - 3$ , putting $x = 3$ , we get, $2{(3)^2} - 7(3) - 3$ . On solving, we get $18 - 21 - 3 = 18 - 24 = - 6$ .
Since it is zero, hence, $x - 3$ is not a factor of the trinomial $2{x^2} - 7x - 3$ .
Therefore, option $(4)$ is correct.

So, the correct answer is “Option (D)”.

Note: Another way to solve this question is ‘Division Method’ i.e., divide each of the given trinomials by $x - 3$ , whose remainder will come out to be $0$ , $x - 3$ will be the factor otherwise not.
The property used in this question is known as ‘Remainder Theorem’.
Pay attention while doing the calculation part, otherwise, the question will go wrong.