Question

Using remainder theorem, find the remainder when $3{{x}^{2}}+x+7$ is divided by x+2
(a)17
(b)12
(c)9
(d)22

Answer 1

Hint: Convert x + 2 in the form of (x – h) and then use the remainder theorem given by “If you divide a polynomial f(x) by (x - h), then the remainder is f(h)” t get the final answer.

To solve the given question we will write the given expression first and assume it as f (x), therefore,

Complete step-by-step answer:

f (x) = $3{{x}^{2}}+x+7$ ……………………………………………………… (1)

As we have to divide the above equation by x + 2 therefore we will assume it as (x – h), therefore we can write,

(x – h) = (x + 2)

As we have to convert (x + 2) in the form of (x – h) and for that we have to replace 2 by [–(– 2)] as both have same values, therefore we will get,

Therefore, (x – h) = [x–(– 2)] ……………………………………. (2)

Now, to proceed further in the solution we should know he remainder theorem given below,
If we divide equation (1) by equation (2) we will get,

\[\therefore \dfrac{f\left( x \right)}{\left( xh \right)}\text{ }=\dfrac{3{{x}^{2}}+x+7}{\left[ x\left( \text{ }2 \right) \right]}\] …………………………………………. (3)

Remainder Theorem:

If you divide a polynomial f(x) by (x - h), then the remainder is f(h).

If we use above theorem in equation (3) we can say that if we divide \[3{{x}^{2}}+x+7\] by [x–(– 2)] then the remainder will be equal to f (– 2) therefore if we put x = -2 in equation (1) we will get,

Therefore, f (– 2) = \[3{{\left( -2 \right)}^{2}}+\left( -2 \right)+7\]

By simplifying the above equation we will get,

Therefore, f (– 2) = \[3\times 4-2+7\]

Further simplification in the above equation will give,

Therefore, f (– 2) = \[12+5\]

Therefore, f (– 2) = 17 …………………………………………… (4)

Therefore by using the remainder theorem the remainder of \[3{{x}^{2}}+x+7\] if we divide it by x + 2 is equal to 17.

Therefore the correct answer is option (a).


Note: You can cross check your answer by long division method if you have time. While using ‘remainder theorem’ do remember to convert the device in the form of (x – h) as we have converted in [x–(– 2)] so that you can avoid confusion.