Question

factoring trinomials : $x^2 + bx + c$
list out the factors to determine the binomials to multiply.
$x^2 - 8x + 16$
pairs of factors | sums of factors
--- | ---
|
look for products of 16 that add up to -8
factored form:
( )( )
check using foil:

Explanation:

Step1: Identify the form

The trinomial is \(x^2 - 8x + 16\), which is a quadratic trinomial of the form \(x^2+bx + c\) (here \(b=-8\), \(c = 16\)). We need to find two numbers that multiply to \(c = 16\) and add up to \(b=-8\).

Step2: List factor pairs of 16

The factor pairs of 16 (considering negative factors since the sum is negative and product is positive) are:

• \((-1,-16)\): Sum is \(-1+(-16)=-17\)
• \((-2,-8)\): Sum is \(-2 + (-8)=-10\)
• \((-4,-4)\): Sum is \(-4+(-4)=-8\)

Step3: Write the factored form

Since the two numbers are \(-4\) and \(-4\), the factored form of \(x^2-8x + 16\) is \((x - 4)(x - 4)\) or \((x - 4)^2\).

Answer:

The factored form is \((x - 4)(x - 4)\) (or \((x - 4)^2\)). For the table:

Pairs of factors	Sums of factors
\((-2,-8)\)	\(-10\)
\((-4,-4)\)	\(-8\)

And the factored form is \((x - 4)(x - 4)\).