To determine the coefficient of [tex]\( x \)[/tex] in the expression [tex]\( (x+3)^2 \)[/tex], let's expand the expression step by step.
1. The expression to expand is [tex]\( (x + 3)^2 \)[/tex].
2. We can use the binomial theorem or simply expand it directly using the distributive property (FOIL method).
Expanding [tex]\( (x + 3)^2 \)[/tex]:
[tex]\[
(x + 3)^2 = (x + 3)(x + 3)
\][/tex]
3. Using the FOIL method (First, Outer, Inner, Last):
[tex]\[
(x + 3)(x + 3) = x \cdot x + x \cdot 3 + 3 \cdot x + 3 \cdot 3
\][/tex]
4. Perform each multiplication:
[tex]\[
x \cdot x = x^2
\][/tex]
[tex]\[
x \cdot 3 = 3x
\][/tex]
[tex]\[
3 \cdot x = 3x
\][/tex]
[tex]\[
3 \cdot 3 = 9
\][/tex]
5. Combine all these terms:
[tex]\[
x^2 + 3x + 3x + 9
\][/tex]
6. Combine like terms (the terms involving [tex]\( x \)[/tex]):
[tex]\[
x^2 + 6x + 9
\][/tex]
7. From the expanded expression [tex]\( x^2 + 6x + 9 \)[/tex], we observe that the coefficient of the term involving [tex]\( x \)[/tex] is [tex]\( 6 \)[/tex].
Thus, the coefficient of [tex]\( x \)[/tex] in [tex]\( (x + 3)^2 \)[/tex] is [tex]\( 6 \)[/tex].
