Question

express your answer as a polynomial in standard form.
$f(x) = x^2 + x + 8$
$g(x) = 3x + 5$
find: $f(g(x))$

Explanation:

Step1: Substitute g(x) into f(x)

We know that \( f(x)=x^{2}+x + 8 \) and \( g(x)=3x + 5 \). To find \( f(g(x)) \), we substitute \( g(x) \) for every \( x \) in \( f(x) \). So we get \( f(g(x))=(3x + 5)^{2}+(3x + 5)+8 \).

Step2: Expand \( (3x + 5)^{2} \)

Using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a = 3x \) and \( b = 5 \), we have \( (3x)^{2}+2\times(3x)\times5+5^{2}=9x^{2}+30x + 25 \).

Step3: Substitute back and simplify

Now substitute the expanded form back into \( f(g(x)) \): \( f(g(x))=9x^{2}+30x + 25+3x + 5+8 \). Combine like terms: \( 9x^{2}+(30x + 3x)+(25 + 5+8)=9x^{2}+33x + 38 \).

Answer:

\( 9x^{2}+33x + 38 \)