<< Chapter < Page Chapter >> Page >
In this section students will:
  • Factor the greatest common factor of a polynomial.
  • Factor a trinomial.
  • Factor by grouping.
  • Factor a perfect square trinomial.
  • Factor a difference of squares.
  • Factor the sum and difference of cubes.
  • Factor expressions using fractional or negative exponents.

Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in [link] .

A large rectangle with smaller squares and a rectangle inside. The length of the outer rectangle is 6x and the width is 10x. The side length of the squares is 4 and the height of the width of the inner rectangle is 4.

The area of the entire region can be found using the formula for the area of a rectangle.

A = l w = 10 x 6 x = 60 x 2  units 2

The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of A = s 2 = 4 2 = 16 units 2 . The other rectangular region has one side of length 10 x 8 and one side of length 4 , giving an area of A = l w = 4 ( 10 x 8 ) = 40 x 32 units 2 . So the region that must be subtracted has an area of 2 ( 16 ) + 40 x 32 = 40 x units 2 .

The area of the region that requires grass seed is found by subtracting 60 x 2 40 x units 2 . This area can also be expressed in factored form as 20 x ( 3 x 2 ) units 2 . We can confirm that this is an equivalent expression by multiplying.

Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions.

Factoring the greatest common factor of a polynomial

When we study fractions, we learn that the greatest common factor    (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, 4 is the GCF of 16 and 20 because it is the largest number that divides evenly into both 16 and 20 The GCF of polynomials works the same way: 4 x is the GCF of 16 x and 20 x 2 because it is the largest polynomial that divides evenly into both 16 x and 20 x 2 .

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

Greatest common factor

The greatest common factor    (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.

Given a polynomial expression, factor out the greatest common factor.

  1. Identify the GCF of the coefficients.
  2. Identify the GCF of the variables.
  3. Combine to find the GCF of the expression.
  4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.
  5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

Factoring the greatest common factor

Factor 6 x 3 y 3 + 45 x 2 y 2 + 21 x y .

First, find the GCF of the expression. The GCF of 6 , 45 , and 21 is 3. The GCF of x 3 , x 2 , and x is x . (Note that the GCF of a set of expressions in the form x n will always be the exponent of lowest degree.) And the GCF of y 3 , y 2 , and y is y . Combine these to find the GCF of the polynomial, 3 x y .

Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that 3 x y ( 2 x 2 y 2 ) = 6 x 3 y 3 , 3 x y ( 15 x y ) = 45 x 2 y 2 , and 3 x y ( 7 ) = 21 x y .

Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.

( 3 x y ) ( 2 x 2 y 2 + 15 x y + 7 )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask