Certainly! Let's find the product of the expressions [tex]\( (Y-3) \)[/tex] and [tex]\( (Y+3) \)[/tex].
### Step-by-Step Solution
1. Write down the expressions to be multiplied:
[tex]\[
(Y-3) \quad \text{and} \quad (Y+3)
\][/tex]
2. Use the distributive property (also known as the FOIL method for binomials) to multiply these two expressions:
[tex]\[
(Y-3)(Y+3)
\][/tex]
3. Applying the distributive property:
[tex]\[
(Y-3)(Y+3) = Y(Y+3) - 3(Y+3)
\][/tex]
4. Multiply each term inside the parentheses:
[tex]\[
Y(Y+3) = Y^2 + 3Y
\][/tex]
[tex]\[
-3(Y+3) = -3Y - 9
\][/tex]
5. Combine these results:
[tex]\[
(Y^2 + 3Y) + (-3Y - 9)
\][/tex]
6. Simplify the expression by combining like terms:
[tex]\[
Y^2 + 3Y - 3Y - 9 = Y^2 - 9
\][/tex]
Therefore, the product of [tex]\( (Y-3) \)[/tex] and [tex]\( (Y+3) \)[/tex] simplifies to [tex]\( Y^2 - 9 \)[/tex].
### Answer:
The product of [tex]\( (Y-3) \)[/tex] and [tex]\( (Y+3) \)[/tex] is:
[tex]\[
Y^2 - 9
\][/tex]
Given the options provided in the question:
- (a) [tex]$Y^4$[/tex]
- (b) [tex]$Y^2+9$[/tex]
- (i) [tex]$Y=(4$[/tex]
- (ii) [tex]$Y+4$[/tex]
None of these options match our result. The correct simplified form is [tex]\( Y^2 - 9 \)[/tex]. Hence, none of the options provided (a), (b), (i), or (ii) are correct as per the standard simplification of the given problem.
