Question

Factor out the greatest common factor from the following polynomial: 28a^7b^7-20ab+16a^4b

Write an expression for the length of the rectangle. (Hint; Factor the area binomial and recall that Area= width • length). Area is 8x^5-32x^2 and the width is 8x^2

Answer 1

To factor out the greatest common factor (GCF) from the polynomial \( 28a^7b^7 - 20ab + 16a^4b \):

1. Identify the coefficients: 28, -20, and 16. The GCF of these numbers is 4.
2. Identify the variables:
- For \( a \), the smallest exponent is 1 (in \( -20ab \)), so we can factor out \( a^1 \).
- For \( b \), the smallest exponent is 1 (also in \( -20ab \)), so we can factor out \( b^1 \).
3. Thus, the GCF is \( 4ab \).

Now factor this GCF out:

\[
28a^7b^7 - 20ab + 16a^4b = 4ab(7a^6b^6 - 5 + 4a^3)
\]

So the factored expression is:

\[
4ab(7a^6b^6 + 4a^3 - 5)
\]

---

Next, to find the length of the rectangle given the area \( 8x^5 - 32x^2 \) and the width \( 8x^2 \):

1. Factor the area first:
\[
8x^5 - 32x^2 = 8x^2(x^3 - 4)
\]
2. We know:
\[
\text{Area} = \text{Width} \times \text{Length}
\]
Thus:
\[
8x^5 - 32x^2 = 8x^2 \times \text{Length}
\]
3. Now set up the equation:
\[
8x^2(x^3 - 4) = 8x^2 \times \text{Length}
\]
4. Divide both sides by \( 8x^2 \):
\[
x^3 - 4 = \text{Length}
\]

So, the expression for the length of the rectangle is:

\[
\text{Length} = x^3 - 4
\]