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Factors, Greatest Common Factor
1 Factors, Greatest Common Factor 6.1
By the end of this section, you should be able to solve the following problems.
1. Find the greatest common factor of :
15m, 12n^{2}, 30p
2. Factor out the greatest common factor and write the
expression in
factored form. Use the distributive property to verify your answer.
9x^{3} − 12x^{5} + 24x^{6}
3. Factor out the “minus” sign and any other common factor.
−3y + 3
4. Factor by grouping.
x^{2}y − x^{2} − 3y + 3
2 Concepts
The greatest common factor among a set of numbers is built
by taking the
product of the prime numbers common to all the numbers. To find the
greatest common factor, we first write the numbers as products of primes.
2.1 Example
Find the greatest common factor (GCF) of 16, 24, and 30
First we break up the numbers into products of their prime factors.
16 = 2 · 2 · 2 · 2
24 = 2 · 2 · 2 · 3
30 = 2 · 3 · 5
From the above factorizations, we see that the only prime
that is common
to all three numbers is 2, and the maximum number of times 2 occurs is
all three lists simultaneously is once , therefore the GCF is 2. In our next
example, we find the GCF among a set of algebraic terms .
2.2 Example
Find the greatest common factor of
12x^{4}, 30x^{3}, 24x^{2}
.
Again, we list all the prime factors of the terms.
12x^{4} = 2· 2 · 3 · x · x · x · x
30x^{3} = 2· 3 · 5 · x · x · x
24x^{2} = 2· 2 · 2 · 3 · x · x
In the lists above, we look for the maximum number of
common factors
that occur simultaneously. Those are,
2 · 3 · x · x .
So the greatest common factor is
6x^{2}
In our next example, we factor out the greatest common
factor and write
the expression in factored form.
2.3 Example
Factor the expression.
8x^{4}y^{3} − 6x^{3}y^{4} + 10x^{2}y^{5}
First we look for the largest number that divides all
three coefficients.
That would be 2. Then we look for the highest power of x that divides
x^{2}, x^{3}, x^{4}, and that would be x^{2}. Finally, the highest power of y that
divides
all three terms is y ^{3}. Writing the expression in factored form we have:
2x^{2}y^{3}(4x^{2} − 3xy + 5y^{3})
An important trick to be able to handle in algebra is to
able to factor out
1 when it may not look like 1 is common factor. But this can always be
done.
2.4 Example
Factor 1 out of the expression.
x −1 = −1 · (1 − x)
We can certainly check this by the distributive law .
−1 · (1 − x) = (−1) · 1 + (−1) · −x = −1 + x = x − 1.
In our next example, we use a technique called factoring
by grouping.
Here we group specific terms together and then find the GCF of just those
grouped terms.
2.5 Example
Factor the expression by grouping.
2x^{2} + 2x − 3x − 3
In the expression above we group the first two terms and
the last two
terms using parentheses .
(2x^{2} + 2x) + (−3x − 3)
Notice that we inserted a plus sign between the grouped
pairs. We never
separate a negative signs from their coefficients. In our next step , we factor
the respective pairs.
2x(x + 1) − 3(x + 1)
Notice the common binomial , (x+1), we will now factor that out.
(x + 1)(2x − 3)
Now the expression is completely factored.
3 Facts
1. The Greatest Common Factor (GCF) is the largest number
or expression
that evenly divides all the numbers or terms.
2. We find the GCF by listing the maximum number of primes
that are
common to all the terms or numbers and then and multiply them together.
3. We can always factor 1 out of any binomial. Look:
a − b = −1 · (b − a)
4. To factor by grouping, group the first two terms of an
expression and
the last two terms of the expression together and insert a plus sign in
between. Never separate a negative sign from its coefficient.
4 Exercises
1. Find the greatest common factor.
15m, 12n^{2}, 30p
2. Factor out the greatest common factor and write the
expression in
factored form. Use the distributive property to verify your answer.
9x^{3} − 12x^{5} + 24x^{5}
3. Factor out the “minus” sign and any other common factor.
−3y + 3 = −3(y − 1)
4. Factor by grouping.
x^{2}y − x^{2} − 3y + 3
5 Solutions
1. Find the greatest common factor.
15m, 12n^{2}, 30p
The greatest common factor is 3.
2. Factor out the greatest common factor and write the
expression in
factored form. Use the distributive property to verify your answer.
9x^{3} − 12x^{5} + 24x^{5}
=
3x^{3}(3 − 4x^{2} + 8^{3})
3. Factor out the “minus” sign and any other common
factor.
−3y + 3 = −3(y − 1)
4. Factor by grouping.
x^{2}y − x^{2} − 3y + 3
=
(x^{2}y − x^{2}) + (−3y + 3)
=
x^{2}(y − 1) − 3(y − 1)
=
(y − 1)(x^{2} − 3)
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