Question

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

Answer 1

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.

Answer 2

Answer: The remainder will be 5 only.

Step-by-step 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)* (3x^3-4x^2+7x-12)+5`

When we compare it with division lemma, which says that

`f(x)=g(x)* q(x)+r(x)`

We get,

`r(x)=5`

Hence, the remainder will be 5 only.