Question

The binomial (x + 5) is a factor of x2 + 8x + 15. What is the other factor?

Answer 1

A

Explanation:

Answer 2

Okay, if
`(x+5)` is a factor of
`x^2+8x+15` then we know that
`x^2+8x+15 = (x+5)( \cdots)` for some other factor which we don't know but want to find out.


`x^2` is the highest power on the left side of the equation so we know we want something in the missing bracket that when we times it by x we get x². We therefore know that we need an x in the missing bracket and so it looks like this
`x^2+8x+15 = (x+5)(x + \cdots)`. When we expand that out we get the x² term (by multiplying the two x's) but also we get a +5x (from multiplying the 5 and the x) but by comparing it to the left hand side again, we see we want it to be +8x, we are +3x short. How can we get +3x?

If we add 3 to the missing bracket we get
`x^2+8x+15 = (x+5)(x + 3)` so we have the
`x^2+5x + \cdots` from the first expansion and now we have
`\cdots + 3x + 15` from the second expansion. We have obtained the extra 3x we need to get to +8x but it has also added on 15 (by multiplying 3 and 5). Luckily, this is also part of the left hand side and sothe two sides match and we have found the missing factor:
`(x+3).`