Question

Which number added to the polynomial \[3x^{2} – 4x – 1\] gives a polynomial with \[(x – 1)\] as a factor?

Answer 1

Hint: In this question, we need to find the number which when added to the polynomial \[3x^{2} – 4x – 1\] gives a polynomial with \[(x – 1)\] as a factor. First, let us consider the given expression as \[f(x)\] . Then by using the factor theorem, we need to find the value of \[x\] from the factor given. Then we need to substitute the value of \[x\] in the given expression . Using this we can find the number which when added to the polynomial \[3x^{2} – 4x – 1\] gives a polynomial with \[(x – 1)\] as a factor.

Complete step-by-step answer:
Given, \[3x^{2} – 4x – 1\]
Let us consider the given expression as \[f(x)\] .
\[\Rightarrow \ f\left( x \right) = 3x^{2} – 4x – 1\]
Also given that \[(x – 1)\] is the factor of the expression \[3x^{2} – 4x – 1\] .
By factor theorem,
If \[(x – 1)\] is the factor of the expression then \[x = 1\] is the
Now on substituting \[x = 1\] ,
We get,
\[\Rightarrow f\left( 1 \right) = 3\left( 1 \right)^{2} – 4\left( 1 \right) – 1\]
Now on simplifying,
We get,
\[\Rightarrow 3 – 4 – 1\]
On further simplifying,
We get,
\[\Rightarrow \ - 2\]
Thus we get when the polynomial \[3x^{2} – 4x – 1\] is divided by \[(x – 1)\] , the remainder is \[- 2\] .
Therefore \[- 2\] is the number added to the polynomial \[3x^{2} – 4x – 1\] gives polynomials with \[(x – 1)\] as a factor.
Final answer :
The number added to the polynomial \[3x^{2} – 4x – 1\] gives polynomial with \[(x – 1)\] as a factor is \[-2\]

Note: In order to solve these types of questions, we should have a strong grip over the factor theorem. Factor theorem states that, let \[f(x)\] be a polynomial equation if \[f(a)\ = 0\] then \[(x – a)\] is the factor of the polynomial \[f(x)\] . Also , In order to reduce the error , we should solve the question step by step wise, as it will help us to solve the question error-free . We can also check whether our answer is correct or not by adding the number \[- 2\] with the given expression.