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

Question

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:

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

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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:

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