1. Home
  2. Maths test
  3. Maths test
  4. Products and factors

Products and factors  (Page 2/4)

<< Chapter < Page
  Maths test     Page 2 / 4
Chapter >> Page >

More products

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

Explain the following terms . (1) Abiotic factors in an ecosystem
Nomai Reply
Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc
Qasim
what is biology
daniel Reply
what is diffusion
Emmanuel Reply
passive process of transport of low-molecular weight material according to its concentration gradient
AI-Robot
what is production?
Catherine
Pathogens and diseases
Reply
how did the oxygen help a human being
Achol Reply
how did the nutrition help the plants
Achol Reply
Biology is a branch of Natural science which deals/About living Organism.
Ahmedin Reply
what is phylogeny
Odigie Reply
evolutionary history and relationship of an organism or group of organisms
AI-Robot
ok
Deng
what is biology
Hajah Reply
cell is the smallest unit of the humanity biologically
Abraham
what is biology
Victoria Reply
what is biology
Abraham
HOW CAN MAN ORGAN FUNCTION
Alfred Reply
the diagram of the digestive system
Assiatu Reply
allimentary cannel
Ogenrwot
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is respiration
Deborah
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Maths test. OpenStax CNX. Feb 09, 2011 Download for free at http://cnx.org/content/col11236/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Maths test' conversation and receive update notifications?

Ask