Question

3x 4 8x 3 14x 2 25x 10 is divided by x, plus, 2x 2.

Answer 1

The polynomial
`\(3x^4 + 8x^3 + 14x^2 + 25x + 10\)` is completely divisible by x + 2, resulting in the quotient
`\(3x^3 - 6x^2 - 4x + 20\)`.

Let's evaluate the synthetic division:

Synthetic division is a method for polynomial division by a linear factor. It simplifies the process by performing division without explicitly writing variables, efficiently obtaining the quotient.

It involves bringing down coefficients, multiplying, adding, and repeating until the last coefficient, providing a quick solution for linear divisors.

Given polynomial:
`\(3x^4 + 8x^3 + 14x^2 + 25x + 10\)`

Divisor: x + 2

Setting up the synthetic division:

`\[\begin{array}cccccc-2 & 3 & 8 & 14 & 25 & 10 \\\end{array}\]`

Now, let's perform the synthetic division:

(check the attached image)

The correct result is
`\(3x^3 - 6x^2 - 4x + 20\)`, and there is no remainder. The polynomial
`\(3x^4 + 8x^3 + 14x^2 + 25x + 10\)` is completely divisible by x + 2.

The complete question is:

Use synthetic division to find the result when 3x^4+8x^3+14x^2+25x+10 is divided by x+2