To find the product of \((5y^2 - 6y)(7y^7 - y^3)\), we will use the distributive property (also known as the FOIL method for two binomials) to multiply each term in the first polynomial by each term in the second polynomial.
Let's perform the multiplication step by step:
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Multiply \(5y^2\) by each term in the second polynomial:
- \(5y^2 \cdot 7y^7 = 35y^{9}\)
- \(5y^2 \cdot (-y^3) = -5y^{5}\)
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Multiply \(-6y\) by each term in the second polynomial:
- \(-6y \cdot 7y^7 = -42y^{8}\)
- \(-6y \cdot (-y^3) = 6y^{4}\)
Now, we can combine all these products:
\[ 35y^9 - 42y^8 - 5y^5 + 6y^4 \]
Next, we will arrange this expression in descending order of exponents:
\[ 35y^9 - 42y^8 - 5y^5 + 6y^4 \]
Therefore, the final polynomial in descending order of exponents is:
\[ 35y^9 - 42y^8 - 5y^5 + 6y^4 \]