Question

What is the remainder when (3x4 + 2x3 − x2 + 2x − 19) ÷ (x + 2)?

Answer 1

Answer: The remainder will be 5 only.

Explanation:

Since we have given that

$f(x)=3x^4+2x^3-x^2+2x-19$

and

$g(x)=x+2$

Now, using the division algorithm, we'll get,

$f(x)=g(x)\times (3x^3-4x^2+7x-12)+5$

When we compare it with  division lemma, which says that

$f(x)=g(x)\times q(x)+r(x)$

We get,

$r(x)=5$

Hence, the remainder will be 5 only.

Answer 2

Answer: Remainder=5

Step-by-step explanation:

We know that the Remainder theorem states that the remainder of the division of a polynomial f(x) by a linear polynomial (x-a) is equal to f(a).

Here $f(x)=3x^4+2x^3-x^2+2x-19$

The linear polynomial = $(x+2)$

$\Rightarrow\ a=-2$

$f(-2)=3(-2)^4+2(-2)^3-(-2)^2+2(-2)-19\\\Rightarrow\ f(-2)=3(16)-16+4-4-19\\\Rightarrow\ f(-2)=48-45\\\Rightarrow\ f(-2)=5$

Hence, the remainder of the given division problem is 5.