Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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.
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.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
