To find the product of [tex]\((y + 3)(y^2 - 3y + 9)\)[/tex], we will use the distributive property by multiplying each term in the binomial by each term in the trinomial. Here is a step-by-step solution:
1. Expand the terms:
We multiply [tex]\( y \)[/tex] by each term of [tex]\((y^2 - 3y + 9)\)[/tex]:
[tex]\[
y \cdot y^2 = y^3
\][/tex]
[tex]\[
y \cdot (-3y) = -3y^2
\][/tex]
[tex]\[
y \cdot 9 = 9y
\][/tex]
2. Next, we multiply [tex]\( 3 \)[/tex] by each term of [tex]\((y^2 - 3y + 9)\)[/tex]:
[tex]\[
3 \cdot y^2 = 3y^2
\][/tex]
[tex]\[
3 \cdot (-3y) = -9y
\][/tex]
[tex]\[
3 \cdot 9 = 27
\][/tex]
3. Combine all the terms:
[tex]\[
y^3 + (-3y^2) + 9y + 3y^2 + (-9y) + 27
\][/tex]
4. Simplify by combining like terms:
- Combine [tex]\( -3y^2 \)[/tex] and [tex]\( 3y^2 \)[/tex]:
[tex]\[
-3y^2 + 3y^2 = 0y^2 = 0
\][/tex]
- Combine [tex]\( 9y \)[/tex] and [tex]\( -9y \)[/tex]:
[tex]\[
9y - 9y = 0y = 0
\][/tex]
5. Write the resulting polynomial:
[tex]\[
y^3 + 0y^2 + 0y + 27 = y^3 + 27
\][/tex]
The correct product is:
[tex]\[
y^3 + 27
\][/tex]
Therefore, the answer is:
[tex]\[
\boxed{y^3 + 27}
\][/tex]
