Clara found the product of [tex]\(3 - 6 y^2\)[/tex] and [tex]\(y^2 + 2\)[/tex]. Is the student's work correct? Her work is shown below:

[tex]\[
\begin{array}{l}
\left(3 - 6 y^2\right)\left(y^2 + 2\right) = 3\left(y^2\right) + \left(-6 y^2\right)(2) \\
= 3 y^2 - 12 y^2 \\
= -9 y^2
\end{array}
\][/tex]

A. No, she did not multiply [tex]\(-6 y^2\)[/tex] by 2 correctly.
B. No, she did not add [tex]\(3 y^2\)[/tex] and [tex]\(-12 y^2\)[/tex] correctly.
C. No, she did not use the distributive property correctly.
D. Yes, she multiplied the binomials correctly.



Answer :

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.