Let's go through the correct steps to multiply the polynomials [tex]\( 3 - 6y^2 \)[/tex] and [tex]\( y^2 + 2 \)[/tex] step-by-step:
1. Write out the polynomials:
[tex]\( (3 - 6y^2) \)[/tex] and [tex]\( (y^2 + 2) \)[/tex].
2. Apply the distributive property (also known as the FOIL method when dealing with binomials):
[tex]\[
(3 - 6y^2)(y^2 + 2)
\][/tex]
We distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[
= 3 \cdot y^2 + 3 \cdot 2 - 6y^2 \cdot y^2 - 6y^2 \cdot 2.
\][/tex]
3. Perform the individual multiplications:
- [tex]\( 3 \cdot y^2 \)[/tex]:
[tex]\[
3y^2
\][/tex]
- [tex]\( 3 \cdot 2 \)[/tex]:
[tex]\[
6
\][/tex]
- [tex]\( -6y^2 \cdot y^2 \)[/tex]:
[tex]\[
-6y^4
\][/tex]
- [tex]\( -6y^2 \cdot 2 \)[/tex]:
[tex]\[
-12y^2
\][/tex]
4. Combine all the terms:
[tex]\[
6 - 12y^2 + 3y^2 - 6y^4
\][/tex]
5. Combine like terms (if any):
The only like terms are [tex]\( -12y^2 \)[/tex] and [tex]\( 3y^2 \)[/tex]:
[tex]\[
6 - 9y^2 - 6y^4
\][/tex]
6. Final polynomial in standard form:
[tex]\[
-6y^4 - 9y^2 + 6
\][/tex]
So, the correctly multiplied polynomial is:
[tex]\[
-6y^4 - 9y^2 + 6
\][/tex]
Now, analyzing the student's work:
- The student's method is incorrect because they did not correctly apply the distributive property to each term in both polynomials and didn't perform all the necessary multiplications.
- They only accounted for one multiplication and missed the cross terms, resulting in an incomplete and incorrect result.
Therefore, the student's work is incorrect as indicated by the following points:
- She did not multiply [tex]\(-6y^2\)[/tex] by [tex]\(2\)[/tex] correctly.
- She did not add the terms correctly (because she failed to consider all necessary terms).
- She did not use the distributive property correctly.
Correct multiplication yields [tex]\(-6y^4 - 9y^2 + 6\)[/tex], not just [tex]\(-9y^2\)[/tex]. So, the correct choice is: No, she did not use the distributive property correctly.
