Grouping

Grouping is also called triple common factoring. It may be a better way to describe this technique.

Polynomials that can be grouped usually have four terms. Two terms can be common factored with one greatest common factor (GCF), and the other two can be common factored with a different GCF. Then, the resulting polynomial will have a binomial GCF that can be common factored a third time.

For example: xy-2x+2y-4

We can see that the first two terms have a common factor of and the second two terms have a common factor of 2: x(y-2)+2(y-2)

Now, there is a (y-2) that can be common factored: (y-2)(x+2)

Be sure to common factor out a negative if the leading coefficient is negative. Otherwise, your two binomials will not be the same: \begin{array}{rl} & 6x^2+3x-14x-7\\ = & 3x(2x+1)-7(2x+1) \\ = & (2x+1)(3x-7) \end{array}

Always make sure that your polynomial is in descending order before grouping: \begin{array}{rl} & 4x+yz+4y+xz \\ = & xz+yz+4x+4y \\ = & z(x+y)+4(x+y) \\ = & (x+y)(z+4) \end{array}

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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