Download Quadratic Relation II

Review
 How do you expand the following:
 3(x+2)  3x  6
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Expand the following:
2( x  4)  2x  8
3(9  x)  27  3xx
3x( x  2)  3 x 2  6 x
5 x(5 x  1)  25 x 2  5 x
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Question for thought
 How do we expand a binomial multiplied by itself?
 How do we expand (x+2)(x+3)?
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EXPANDING BINOMIALS
Unit – Quadratics Relations II
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Key Definitions
 Binomial – an algebraic expression containing two
unlike terms (i.e. x+3, 2x+y, 3x2-y)
 Trinomial -A polynomial with three terms
 Polynomial – an algebraic expression made up of
terms that are added and subtracted.
( i.e 8x + 3y – 5x + 6y)
 Quadratic equation – an expression in the form
ax2+bx+c, where a≠0.
 Term - a number or a variable, or the product of
numbers and variables (i.e. 2, 2x, 2xy 2x2yz)
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 Distributive Property – multiply the terms i.e.
a ( b + c ) = ab + ac
e.g 2 (3y + 8) = 2x3y + 2x8
Binomial Expansion using
Distributive Property
 Method 1:
 Think of x+2 as being a number:
( x  2)( x  3)
 x( x  2)  3( x  2)
 x 2  2 x  3x  6
 Collect Like terms
* very important
 x 2  5x  6
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Binomial Expansion
Outside
 Method 2: FOIL
 First
 Outside
 Inside
First
( x  2)( x  3)
 Last
Inside
Last
 x  3x  2 x  6
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Binomial Expansion
 Method 2: FOIL
 First
 Outside
 Inside
 Last
( x  2)( x  3)
 x  5x  6
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(x – 2) (x + 4) FOIL METHOD
x2 + 4x – 2x – 8
x2 + 2x – 8
Try it yourself
 Expand the following:
( x  1)( x  2) 
( x  4)( x  5) 
( x  3)( x  1) 
x( x  1)( x  2) 
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Try it yourself
 Expand the following:
( x  1)( x  2)  x  3x  2
2
( x  4)( x  5)  x  x  20
2
( x  3)( x  1)  x  4 x  3
2
x( x  1)( x  2)  x( x  x  2)
2
x( x  1)( x  2)  x  x  2 x
3
2
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Two Special Products
 There are two special cases of binomial products which DO NOT
require the use of the distributive property or FOIL to simplify.
Case 1. Squaring a Binomial
Expand and simplify
(x – 2)2
(x – 2)(x – 2)
x2 – 2x – 2x + 4
x2 – 4x + 4
(4x + 3)2
(4x + 3)(4x + 3)
16x2 + 12x + 12x + 9
16x2 + 24x + 9
 In general:
( a + b)2 = a2 + 2ab + b2
1st term squared
Always
2 x 1st term x 2nd term
2nd term squared
 Case 2. Difference of Squares
Expand and simplify
(x + 3)(x – 3)
x2 – 3x + 3x – 9
x2 – 9
(4x + 2)(4x – 2)
16x2 – 8x + 8x – 4
16x2 – 4
 In general:
( a + b) (a-b) = a2 - b2
1st term squared
Always minus
2nd term squared
1. (x + 4)2
x2 + 8x + 16
2. (x – 5)(x + 5)
x2 – 25
3. (2x + 3)(2x – 3)
4x2 – 9
4. (3x – 7)2
9x2 -42x + 49