Mixed Signed Factorising

Question

I need hep factoring mixed signed expressions. I know how to factorise but I'm getting really confused on to which side to put the negetive sign on etc.

E.G.

$1)$ $x^2 - 10x + 16$

My Steps:

$1)$ I put the $x's$ in the brackets first.

$2)$ Then I found out two numbers which multiply to make the last expression and add to get the middle expression. In this instance I got $2$ and $8$.

$3)$ So putting thing into the brackets, I got - $(x + 2)(x - 8)$

No matter how many times I changed the signs or swapped them over or the numbers I didn't get the right answer. Does anyone know how not to get confused with signs and how come I can't work out this question?

Answer 1

In your expression $x^2-10x+16$, you have the coefficient of $x$ as $-10$ which is negative and the constant as $16$ which is positive.

You are right about there being $8$ and $2$ involved, so your factorisation will be

$x^2-10x+16=(x*2)(x**8) = x^2+((*2)+(**8))x+((*2)\times(**8))$

where symbols $*$ and $**$ represent either a minus or a plus sign.

So you have

$(*2)\times(**8)=+16$

what does the above tell you about $*$ and $**$? Should they be the same or different?

and

$(*2) + (**8)=-10$

Now, what do you think $*$ and $**$ will be?

Answer 2

$$x^2-10x+16=x^2-(8+2)x+8\cdot2=x^2-8x-2x+8\cdot2$$ $$=x(x-8)-2(x-8)=(x-8)(x-2)$$