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( x + 5 ) ( x 5 ) = x 2 25 ( x + 11 ) ( x 11 ) = x 2 121 ( 2 x + 3 ) ( 2 x 3 ) = 4 x 2 9

Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.

Is there a special form for the sum of squares?

No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares.

Difference of squares

When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.

( a + b ) ( a b ) = a 2 b 2

Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares.

  1. Square the first term of the binomials.
  2. Square the last term of the binomials.
  3. Subtract the square of the last term from the square of the first term.

Multiplying binomials resulting in a difference of squares

Multiply ( 9 x + 4 ) ( 9 x 4 ) .

Square the first term to get ( 9 x ) 2 = 81 x 2 . Square the last term to get 4 2 = 16. Subtract the square of the last term from the square of the first term to find the product of 81 x 2 16.

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Multiply ( 2 x + 7 ) ( 2 x 7 ) .

4 x 2 −49

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Performing operations with polynomials of several variables

We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:

( a + 2 b ) ( 4 a b c ) a ( 4 a b c ) + 2 b ( 4 a b c ) Use the distributive property . 4 a 2 a b a c + 8 a b 2 b 2 2 b c Multiply . 4 a 2 + ( a b + 8 a b ) a c 2 b 2 2 b c Combine like terms . 4 a 2 + 7 a b a c 2 b c 2 b 2 Simplify .

Multiplying polynomials containing several variables

Multiply ( x + 4 ) ( 3 x 2 y + 5 ) .

Follow the same steps that we used to multiply polynomials containing only one variable.

x ( 3 x 2 y + 5 ) + 4 ( 3 x 2 y + 5 )   Use the distributive property . 3 x 2 2 x y + 5 x + 12 x 8 y + 20 Multiply . 3 x 2 2 x y + ( 5 x + 12 x ) 8 y + 20 Combine like terms . 3 x 2 2 x y + 17 x 8 y + 20   Simplify .
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Multiply ( 3 x 1 ) ( 2 x + 7 y 9 ) .

6 x 2 + 21 x y −29 x −7 y + 9

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Access these online resources for additional instruction and practice with polynomials.

Key equations

perfect square trinomial ( x + a ) 2 = ( x + a ) ( x + a ) = x 2 + 2 a x + a 2
difference of squares ( a + b ) ( a b ) = a 2 b 2

Key concepts

  • A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See [link] .
  • We can add and subtract polynomials by combining like terms. See [link] and [link] .
  • To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See [link] .
  • FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See [link] .
  • Perfect square trinomials and difference of squares are special products. See [link] and [link] .
  • Follow the same rules to work with polynomials containing several variables. See [link] .

Section exercises

Verbal

Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.

The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.

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Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial.

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You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?

Use the distributive property, multiply, combine like terms, and simplify.

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State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial.

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Algebraic

For the following exercises, identify the degree of the polynomial.

14 m 3 + m 2 16 m + 8

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200 p 30 p 2 m + 40 m 3

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6 y 4 y 5 + 3 y 4

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For the following exercises, find the sum or difference.

( 12 x 2 + 3 x ) ( 8 x 2 −19 )

4 x 2 + 3 x + 19

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( 4 z 3 + 8 z 2 z ) + ( −2 z 2 + z + 6 )

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( 6 w 2 + 24 w + 24 ) ( 3 w 2 6 w + 3 )

3 w 2 + 30 w + 21

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( 7 a 3 + 6 a 2 4 a 13 ) + ( 3 a 3 4 a 2 + 6 a + 17 )

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( 11 b 4 6 b 3 + 18 b 2 4 b + 8 ) ( 3 b 3 + 6 b 2 + 3 b )

11 b 4 −9 b 3 + 12 b 2 −7 b + 8

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( 49 p 2 25 ) + ( 16 p 4 32 p 2 + 16 )

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For the following exercises, find the product.

( 4 x + 2 ) ( 6 x 4 )

24 x 2 −4 x −8

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( 14 c 2 + 4 c ) ( 2 c 2 3 c )

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( 6 b 2 6 ) ( 4 b 2 4 )

24 b 4 −48 b 2 + 24

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( 3 d 5 ) ( 2 d + 9 )

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( 9 v 11 ) ( 11 v 9 )

99 v 2 −202 v + 99

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( 4 t 2 + 7 t ) ( −3 t 2 + 4 )

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( 8 n 4 ) ( n 2 + 9 )

8 n 3 −4 n 2 + 72 n −36

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For the following exercises, expand the binomial.

( 3 y 7 ) 2

9 y 2 −42 y + 49

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( 4 p + 9 ) 2

16 p 2 + 72 p + 81

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( 3 y 6 ) 2

9 y 2 −36 y + 36

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For the following exercises, multiply the binomials.

( 4 c + 1 ) ( 4 c 1 )

16 c 2 −1

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( 9 a 4 ) ( 9 a + 4 )

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( 15 n 6 ) ( 15 n + 6 )

225 n 2 −36

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( 25 b + 2 ) ( 25 b 2 )

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( 4 + 4 m ) ( 4 4 m )

−16 m 2 + 16

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( 14 p + 7 ) ( 14 p 7 )

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( 11 q 10 ) ( 11 q + 10 )

121 q 2 −100

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For the following exercises, multiply the polynomials.

( 2 x 2 + 2 x + 1 ) ( 4 x 1 )

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( 4 t 2 + t 7 ) ( 4 t 2 1 )

16 t 4 + 4 t 3 −32 t 2 t + 7

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( x 1 ) ( x 2 2 x + 1 )

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( y 2 ) ( y 2 4 y 9 )

y 3 −6 y 2 y + 18

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( 6 k 5 ) ( 6 k 2 + 5 k 1 )

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( 3 p 2 + 2 p 10 ) ( p 1 )

3 p 3 p 2 −12 p + 10

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( 4 m 13 ) ( 2 m 2 7 m + 9 )

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( a + b ) ( a b )

a 2 b 2

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( 4 x 6 y ) ( 6 x 4 y )

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( 4 t 5 u ) 2

16 t 2 −40 t u + 25 u 2

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( 9 m + 4 n 1 ) ( 2 m + 8 )

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( 4 t x ) ( t x + 1 )

4 t 2 + x 2 + 4 t −5 t x x

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( b 2 1 ) ( a 2 + 2 a b + b 2 )

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( 4 r d ) ( 6 r + 7 d )

24 r 2 + 22 r d −7 d 2

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( x + y ) ( x 2 x y + y 2 )

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Real-world applications

A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: ( 4 x + 1 ) ( 8 x 3 ) where x is measured in meters. Multiply the binomials to find the area of the plot in standard form.

32 x 2 −4 x −3 m 2

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A prospective buyer wants to know how much grain a specific silo can hold. The area of the floor of the silo is ( 2 x + 9 ) 2 . The height of the silo is 10 x + 10 , where x is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold.

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Extensions

For the following exercises, perform the given operations.

( 4 t 7 ) 2 ( 2 t + 1 ) ( 4 t 2 + 2 t + 11 )

32 t 3 100 t 2 + 40 t + 38

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( 3 b + 6 ) ( 3 b 6 ) ( 9 b 2 36 )

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( a 2 + 4 a c + 4 c 2 ) ( a 2 4 c 2 )

a 4 + 4 a 3 c −16 a c 3 −16 c 4

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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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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
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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
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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
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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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