Question

factor the polynomial completely using the x method.\\(x^2 + 13x - 48\\)\
what is the four - term polynomial and factored form of the polynomial?\
\\(\circ\\ x^2 + 7x + 6x - 48=(x + 7)(x + 6)\\)\
\\(\circ\\ x^2 - 12x + 4x - 48=(x - 12)(x + 4)\\)\
\\(\circ\\ x^2 + 16x - 3x - 48=(x + 16)(x - 3)\\)\
\\(\circ\\ x^2 - 16x + 3x - 48=(x - 16)(x + 3)\\)

Explanation:

Step1: Identify $ac$ and $b$

For $x^2+13x-48$, $a=1$, $c=-48$, so $ac = 1\times(-48) = -48$; $b=13$.

Step2: Find pair for $ac$ and $b$

Find two numbers that multiply to $-48$ and add to $13$: $16$ and $-3$ (since $16\times(-3)=-48$, $16+(-3)=13$).

Step3: Rewrite as four-term polynomial

Split the middle term using the pair: $x^2+16x-3x-48$

Step4: Factor by grouping

Group and factor:

$$\begin{align*} (x^2+16x)+(-3x-48)&=x(x+16)-3(x+16)\\ &=(x+16)(x-3) \end{align*}$$

Answer:

$\boldsymbol{x^2 + 16x - 3x - 48 = (x + 16)(x - 3)}$