Sure, let's find the product of the polynomials [tex]\((6 - y - 4y^2)\)[/tex] and [tex]\((-5 + 7y^2)\)[/tex] and express the result in standard form.
Given:
[tex]\[
(6 - y - 4y^2)(-5 + 7y^2)
\][/tex]
First, we will distribute each term of the first polynomial with each term of the second polynomial. This is commonly known as the distributive property (or FOIL method in the case of binomials).
Let's distribute each term:
1. [tex]\(6 \cdot (-5 + 7y^2)\)[/tex]
2. [tex]\(-y \cdot (-5 + 7y^2)\)[/tex]
3. [tex]\(-4y^2 \cdot (-5 + 7y^2)\)[/tex]
Now, perform the distribution:
1. [tex]\(6 \cdot -5 = -30\)[/tex]
2. [tex]\(6 \cdot 7y^2 = 42y^2\)[/tex]
Combining these results, we get:
[tex]\[6 \cdot (-5 + 7y^2) = -30 + 42y^2\][/tex]
Next, for the second term:
[tex]\[ -y \cdot -5 = 5y \][/tex]
[tex]\[ -y \cdot 7y^2 = -7y^3 \][/tex]
Combining these results, we get:
[tex]\[ -y \cdot (-5 + 7y^2) = 5y - 7y^3 \][/tex]
Finally, for the third term:
[tex]\[ -4y^2 \cdot -5 = 20y^2 \][/tex]
[tex]\[ -4y^2 \cdot 7y^2 = -28y^4 \][/tex]
Combining these, we get:
[tex]\[ -4y^2 \cdot (-5 + 7y^2) = 20y^2 - 28y^4 \][/tex]
Now, combine all the terms:
[tex]\[
-30 + 42y^2 + 5y - 7y^3 + 20y^2 - 28y^4
\][/tex]
Group like terms:
[tex]\[
-30 + (42y^2 + 20y^2) + 5y - 7y^3 - 28y^4
\][/tex]
Simplify the coefficients:
[tex]\[
-30 + 62y^2 + 5y - 7y^3 - 28y^4
\][/tex]
Rewriting this in standard form (order by powers of [tex]\(y\)[/tex] from highest to lowest):
[tex]\[
-28y^4 - 7y^3 + 62y^2 + 5y - 30
\][/tex]
So, the standard form of the polynomial that represents the product is:
[tex]\[
\boxed{-28y^4 - 7y^3 + 62y^2 + 5y - 30}
\][/tex]
