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Question 2, A.2.5

Subtract the polynomials.

[tex]\[
\left(7y^3 + 4y^2 - y - 9\right) - \left(y^2 - 9y + 4\right)
\][/tex]

The difference is [tex]\(\square\)[/tex]. (Simplify your answer.)

Sagot :

GinnyAnsweridn
Certainly! Let's go through the step-by-step process to subtract the given polynomials:

Given polynomials:
[tex]\[ P(y) = 7y^3 + 4y^2 - y - 9 \][/tex]
[tex]\[ Q(y) = y^2 - 9y + 4 \][/tex]

We need to subtract [tex]\( Q(y) \)[/tex] from [tex]\( P(y) \)[/tex].

[tex]\[ (P(y) - Q(y)) = (7y^3 + 4y^2 - y - 9) - (y^2 - 9y + 4) \][/tex]

To perform the subtraction, distribute the negative sign through [tex]\( Q(y) \)[/tex]:

[tex]\[ = 7y^3 + 4y^2 - y - 9 - y^2 + 9y - 4 \][/tex]

Now, let's combine like terms:

1. [tex]\( y^3 \)[/tex] term:
- There is only one [tex]\( y^3 \)[/tex] term:
[tex]\[ 7y^3 \][/tex]

2. [tex]\( y^2 \)[/tex] terms:
- Combine [tex]\( 4y^2 \)[/tex] and [tex]\( -y^2 \)[/tex]:
[tex]\[ 4y^2 - y^2 = 3y^2 \][/tex]

3. [tex]\( y \)[/tex] terms:
- Combine [tex]\( -y \)[/tex] and [tex]\( 9y \)[/tex]:
[tex]\[ -y + 9y = 8y \][/tex]

4. Constant terms:
- Combine [tex]\( -9 \)[/tex] and [tex]\( -4 \)[/tex]:
[tex]\[ -9 - 4 = -13 \][/tex]

Putting all these together, the expression simplifies to:

[tex]\[ 7y^3 + 3y^2 + 8y - 13 \][/tex]

Thus, the difference is [tex]\( 7y^3 + 3y^2 + 8y - 13 \)[/tex].
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