Question

which of the binomials below is a factor of this trinomial? x^2+6x+8

Answer 1

where?

Answer 2

(x+4)(x+2), lol which one of those do you have?

Answer 3

Here, why don't you just learn how to factor it?

Multiplying two binomials will give us either a binomial or a trinomial. \[(x+a)(x+b) = x(x+b)+a(x+b) = x^2 + bx + ax + ab\]\[(x+a)(x+b) = x^2 + (a+b)x + ab\]If \(a = -b\) then we get \[x^2 -ab\]but otherwise we have our trinomial.

To factor, we want to go the opposite direction: we need to undo those two applications of the distributive property and find the original product binomials. Let's compare our trinomial with the one I multiplied out:

\[x^2+6x+8\]\[x^2+(a+b)x + ab\]We can see that those will be equal if (and only if) \[6x = (a+b)x\implies 6 = a+b\]and\[8 = ab\]

So, to factor, we look at all the pairs of factors which will give us 8 if multiplied, and find the pair that gives us 6 if added.

\[1*8 = 8, 1+8 = 9\]\[2*4 = 8, 2+4 = 6\]\[4*2 = 8, 4+2=6\]\[8*1 = 8, 8+1 = 9\]\[-1*-8 = 8, -1+-8 = -9\]\[-2*-4 = 8, -2+-4 = -6\]\[-4*-2 = 8, -4+-2 = -6\]\[-8*-1 = 8, -8 + -1 = -9\]

As you can see, the only pair of factors that works is \(4,2\) or \(2,4\). Plugging those values into our equation:
\[(x+4)(x+2) = x(x+2) + 4(x+2) = x*x + 2*x +4*x + 4*2\]\[(x+4)(x+2) = x^2 + 2x + 4x + 8\]\[(x+4)(x+2) = x^2 +6x + 8\checkmark\]