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Khan academy video on products of polynomials.

We have seen how to multiply two binomials in "Product of Two Binomials" . In this section, we learn how to multiply a binomial (expression with two terms) by a trinomial (expression withthree terms). Fortunately, we use the same methods we used to multiply two binomials to multiply a binomial and a trinomial.

For example, multiply 2 x + 1 by x 2 + 2 x + 1 .

( 2 x + 1 ) ( x 2 + 2 x + 1 ) = 2 x ( x 2 + 2 x + 1 ) + 1 ( x 2 + 2 x + 1 ) ( apply distributive law ) = [ 2 x ( x 2 ) + 2 x ( 2 x ) + 2 x ( 1 ) ] + [ 1 ( x 2 ) + 1 ( 2 x ) + 1 ( 1 ) ] = 2 x 3 + 4 x 2 + 2 x + x 2 + 2 x + 1 ( expand the brackets ) = 2 x 3 + ( 4 x 2 + x 2 ) + ( 2 x + 2 x ) + 1 ( group like terms to simplify ) = 2 x 3 + 5 x 2 + 4 x + 1 ( simplify to get final answer )
Multiplication of Binomial with Trinomial

If the binomial is A + B and the trinomial is C + D + E , then the very first step is to apply the distributive law:

( A + B ) ( C + D + E ) = A ( C + D + E ) + B ( C + D + E )

If you remember this, you will never go wrong!

Multiply x - 1 with x 2 - 2 x + 1 .

  1. We are given two expressions: a binomial, x - 1 , and a trinomial, x 2 - 2 x + 1 . We need to multiply them together.

  2. Apply the distributive law and then simplify the resulting expression.

  3. ( x - 1 ) ( x 2 - 2 x + 1 ) = x ( x 2 - 2 x + 1 ) - 1 ( x 2 - 2 x + 1 ) ( apply distributive law ) = [ x ( x 2 ) + x ( - 2 x ) + x ( 1 ) ] + [ - 1 ( x 2 ) - 1 ( - 2 x ) - 1 ( 1 ) ] = x 3 - 2 x 2 + x - x 2 + 2 x - 1 ( expand the brackets ) = x 3 + ( - 2 x 2 - x 2 ) + ( x + 2 x ) - 1 ( group like terms to simplify ) = x 3 - 3 x 2 + 3 x - 1 ( simplify to get final answer )
  4. The product of x - 1 and x 2 - 2 x + 1 is x 3 - 3 x 2 + 3 x - 1 .

Find the product of x + y and x 2 - x y + y 2 .

  1. We are given two expressions: a binomial, x + y , and a trinomial, x 2 - x y + y 2 . We need to multiply them together.

  2. Apply the distributive law and then simplify the resulting expression.

  3. ( x + y ) ( x 2 - x y + y 2 ) = x ( x 2 - x y + y 2 ) + y ( x 2 - x y + y 2 ) ( apply distributive law ) = [ x ( x 2 ) + x ( - x y ) + x ( y 2 ) ] + [ y ( x 2 ) + y ( - x y ) + y ( y 2 ) ] = x 3 - x 2 y + x y 2 + y x 2 - x y 2 + y 3 ( expand the brackets ) = x 3 + ( - x 2 y + y x 2 ) + ( x y 2 - x y 2 ) + y 3 ( group like terms to simplify ) = x 3 + y 3 ( simplify to get final answer )
  4. The product of x + y and x 2 - x y + y 2 is x 3 + y 3 .

We have seen that:
( x + y ) ( x 2 - x y + y 2 ) = x 3 + y 3

This is known as a sum of cubes .

Investigation : difference of cubes

Show that the difference of cubes ( x 3 - y 3 ) is given by the product of x - y and x 2 + x y + y 2 .

Products

  1. Find the products of:
    (a) ( - 2 y 2 - 4 y + 11 ) ( 5 y - 12 ) (b) ( - 11 y + 3 ) ( - 10 y 2 - 7 y - 9 )
    (c) ( 4 y 2 + 12 y + 10 ) ( - 9 y 2 + 8 y + 2 ) (d) ( 7 y 2 - 6 y - 8 ) ( - 2 y + 2 )
    (e) ( 10 y 5 + 3 ) ( - 2 y 2 - 11 y + 2 ) (f) ( - 12 y - 3 ) ( 12 y 2 - 11 y + 3 )
    (g) ( - 10 ) ( 2 y 2 + 8 y + 3 ) (h) ( 2 y 6 + 3 y 5 ) ( - 5 y - 12 )
    (i) ( 6 y 7 - 8 y 2 + 7 ) ( - 4 y - 3 ) ( - 6 y 2 - 7 y - 11 ) (j) ( - 9 y 2 + 11 y + 2 ) ( 8 y 2 + 6 y - 7 )
    (k) ( 8 y 5 + 3 y 4 + 2 y 3 ) ( 5 y + 10 ) ( 12 y 2 + 6 y + 6 ) (l) ( - 7 y + 11 ) ( - 12 y + 3 )
    (m) ( 4 y 3 + 5 y 2 - 12 y ) ( - 12 y - 2 ) ( 7 y 2 - 9 y + 12 ) (n) ( 7 y + 3 ) ( 7 y 2 + 3 y + 10 )
    (o) ( 9 ) ( 8 y 2 - 2 y + 3 ) (p) ( - 12 y + 12 ) ( 4 y 2 - 11 y + 11 )
    (q) ( - 6 y 4 + 11 y 2 + 3 y ) ( 10 y + 4 ) ( 4 y - 4 ) (r) ( - 3 y 6 - 6 y 3 ) ( 11 y - 6 ) ( 10 y - 10 )
    (s) ( - 11 y 5 + 11 y 4 + 11 ) ( 9 y 3 - 7 y 2 - 4 y + 6 ) (t) ( - 3 y + 8 ) ( - 4 y 3 + 8 y 2 - 2 y + 12 )


Factorising a quadratic

Khan academy video on factorising a quadratic.

Factorisation can be seen as the reverse of calculating the product of factors. In order to factorise a quadratic, we need to find the factors which when multiplied together equal the original quadratic.

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?
Aislinn Reply
cm
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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
Jude Reply
Can you compute that for me. Ty
Jude
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emma Reply
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what is inorganic
emma
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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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
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
Krampah Reply
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.
Sahid Reply
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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answer
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progressive wave
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Mujahid
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?
yasuo Reply
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2
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