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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 )
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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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