Question

If x+a is a factor of 2x²+2ax+5x+10,find a.

Answer 1

If
`x+a` is a factor of the polynomial, it means that
`-a` is a root of the polynomial.

In fact, if a polynomial
`p(x)` has a root
`x_0` (i.e.
`p(x_0)=0`), then the polynomial is divisible by
`(x-x_0)`

In your case, you know that the polynomial is divisible by
`(x+a)=(x-(-a))`, so
`-a` is a solution.

Let's evaluate
`p(-a)` and set it to zero:

`2(-a)^2+2a(-a)+5(-a)+10 = 0 \iff 2a^2-2a^2-5a+10=0 \iff -5a+10 = 0 \iff a = 2`

We can check the answer: if
`a=2` the polynomial becomes

`2x^2+2(2)x+5x+10 = 2x^2+9x+10`

And we're claiming that
`-a=-2` is a root of this polynomial:

`2(-2)^2+9(-2)+10 = 2\cdot 4 - 18 + 10 = 0`