Common Factoring

Common factoring involves separating the greatest common factor from each term of a polynomial by “pulling it out”.

We don’t common factor if the GCF is 1 because that would not be changing the expression.

We separate the GCF from the polynomial using division. The GCF is written first, then in brackets, each term of the polynomial is divided by the GCF. It’s the opposite of distributive property.

When factoring, you should always common factor first!.

We can common factor the GCF of 4x^2 from the expression 20x^3-8x^220x^3-8x^2=4x^2\Big(\frac{20x^3}{4x^2}-\frac{8x^2}{4x^2}\Big)=4x^2(5x-2)

You can check if you’ve factored correctly by expanding and simplifying: 4x^2(5x-2)=20x^3-8x^2

If the leading coefficient is negative, common factor out a negative coefficient: -8x^3y^2+20x^2y^2-16x^2y=-4x^2y(2xy-5y+4)

Remember that any polynomial divided by itself is 1! 35x^3y^2-28xy^2+7xy=7xy(5x^2-4y+1)

Even polynomials may be a part of the greatest common factor if they are contained in brackets 2x(x-3)-5(x-3)=(x-3)(2x-5)

Practice

EXAMPLE 1

Simplify then evaluate: 2^{11} div 2^8

SOLUTION

Using the Law of Division: 2^{11} div 2^8=2^{11-8}=2^3

Evaluating: 2^3=8

EXAMPLE 2

Simplify: (m^0)^{10}

SOLUTION

Using the Law of Powers: (m^0)^{10}=m^0

We should more properly use the Law of Zero Exponents to write: m^0=1

EXAMPLE 3

Find the value of x that makes the equation true: 4^x times 4^4=4^7

SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: x+4=7

Therefore: x=3

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