To find the product of \( (5y^2 - 6y)(7y^7 - y^3) \), we can apply the distributive property (also known as the FOIL method for binomials), which means we multiply each term in the first polynomial by each term in the second polynomial.
Let's break this down step-by-step:
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First, distribute \( 5y^2 \):
- \( 5y^2 \cdot 7y^7 = 35y^{2+7} = 35y^9 \)
- \( 5y^2 \cdot (-y^3) = -5y^{2+3} = -5y^5 \)
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Second, distribute \( -6y \):
- \( -6y \cdot 7y^7 = -42y^{1+7} = -42y^8 \)
- \( -6y \cdot (-y^3) = 6y^{1+3} = 6y^4 \)
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Now combine all these products:
- From \( 5y^2 \): \( 35y^9, -5y^5 \)
- From \( -6y \): \( -42y^8, 6y^4 \)
Combining these we get:
\[ 35y^9 - 42y^8 - 5y^5 + 6y^4 \]
- Now, write the final answer in descending order of exponents:
- The final expression is:
\[ \boxed{35y^9 - 42y^8 - 5y^5 + 6y^4} \]
This should give you the product with all terms properly combined and ordered by exponent.
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