Question

given:
$f(x) = x^2 - 5$
$g(x) = x^2 + 5$
what is $f(x) \cdot g(x)$?
a. $2x^2$
b. $x^4 - 25$
c. $x^4 - 10x^2 + 30$
d. $x^4 + 10x^2 + 20$

Explanation:

Step1: Recall the formula for multiplying two binomials (difference of squares)

We know that \((a - b)(a + b)=a^{2}-b^{2}\). Here, \(f(x)=x^{2}-5\) and \(g(x)=x^{2}+5\), so \(f(x)\cdot g(x)=(x^{2}-5)(x^{2}+5)\).

Step2: Apply the difference of squares formula

Let \(a = x^{2}\) and \(b = 5\). Then \((x^{2}-5)(x^{2}+5)=(x^{2})^{2}-5^{2}\).

Step3: Simplify the exponents and the constant term

\((x^{2})^{2}=x^{4}\) and \(5^{2} = 25\), so \((x^{2})^{2}-5^{2}=x^{4}-25\).

Answer:

B. \(x^{4}-25\)