Question

cphs : advanced algebra: concepts and connections - block (27.0831030)
adding and subtracting rational expressions
factor the polynomial:
$x^2 + 5x - 14 =$ a $(x - 2) \times$ b $(x + 7)$
$x^2 - 10x + 16 =$ c $(x - 2) \times$ d $(x - 8)$
this is a multi - part item.
which polynomial is a factor of both expressions?
options: $x - 8$, (blank), $(x - 2)^2$, $x + 7$, $x - 2$

Explanation:

Step1: Analyze the first polynomial

We have the polynomial \(x^{2}+5x - 14\). To factor it, we need two numbers that multiply to \(- 14\) and add up to \(5\). The numbers are \(7\) and \(-2\) since \(7\times(-2)=-14\) and \(7+( - 2)=5\). So, \(x^{2}+5x - 14=(x + 7)(x-2)\).

Step2: Analyze the second polynomial

We have the polynomial \(x^{2}-10x + 16\). To factor it, we need two numbers that multiply to \(16\) and add up to \(-10\). The numbers are \(-2\) and \(-8\) since \((-2)\times(-8)=16\) and \((-2)+(-8)=-10\). So, \(x^{2}-10x + 16=(x - 2)(x - 8)\).

Step3: Find the common factor

From the factorizations, the first polynomial factors to \((x + 7)(x-2)\) and the second factors to \((x - 2)(x - 8)\). The common factor is \((x - 2)\).

Answer:

\(x - 2\) (or in the boxed form as per the options, the option with \(x - 2\))