Multiplication of Polynomials

Report
Section 5.2
Multiplying Polynomials
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Multiplying Two Monomials
Multiplying a Polynomial
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By a number
By a monomial
By another polynomial
The FOIL Method
Multiplying 3 or More Polynomials
Special Products
Simplifying Expressions
Applications
5.2
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Multiplying Two (or more) Monomials
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Multiply the numeric coefficients
Add exponents of matched variables
Include any unmatched variables
Learn to
do these
IN YOUR
HEAD!
Do the variables
Examples
in alpha order
(3)(2x) = 6x -4y(-2xy) = 8xy2 -2s(r) = -2rs
3x(2x)(3x) = 18x3
-5x3(4x2y) = -20x5y
-2(-y) = 2y
(-2b3)(3a)(a2bc) = -6a3b4c
5.2
2
For You
(  8 x y )( 5 x y )
4
7
 40 x y
7
3
2
(  2 x yz )(  6 x y z )
2
9
7
12 x y
5.2
5
11
5
z
10
2
7
3
Multiplying a Polynomial by a Number
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Positive numbers – law of distribution
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5 times 2x2 – 3x + 7
5(2x2) – 5(3x) + 5(7)
10x2 – 15x + 35
Do this in your head?
Negative numbers – be careful!
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-3 times 4y3 – 6y2 + y – 2
-3(4y3)– -3(6y2)+ -3(y) - -3(2)
-12y3 + 18y2 – 3y + 6
5.2
In your head?
4
Multiplying a polynomial by a
monomial
To multiply a polynomial by a monomial, we multiply each
term of the polynomial by the monomial.
3x2(6xy + 3y2) = 18x3y + 9x2y2
5x3y2(xy3 – 2x2y) = 5x4y5 – 10x5y3
-2ab2(3bz – 2az + 4z3) = -6ab3z + 4a2b2z – 8ab2z3
5.2
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Multiplying a Polynomial by a
Polynomial (in general)
To multiply a polynomial by a polynomial, we
use the distributive property repeatedly.
Horizontal Method:
(2a + b)(3a – 2b) = 2a(3a – 2b) + b(3a – 2b)
= 6a2 – 4ab + 3ab – 2b2 = 6a2 –ab – 2b2
Vertical Method: 3x2 + 2x – 5
4x + 2
6x2 + 4x – 10
12x3 + 8x2 – 20x____
12x3 + 14x2 – 16x – 10
5.2
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Bigger Multiplications
Leave Missing
Variable Space
( 5 x  x  4 )(  2 x  3 x  6 )
3
2
5x
Leave
Margin
Space

30 x
15 x
  10 x
 2x  3x  6
2
 6 x  24
3
 3 x  12 x
4
2
 2x
5
x4
3
 8x
3
2
 10 x  15 x  28 x  11 x  6 x  24
5
4
3
2
5.2
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FOIL: Used to Multiply Two Binomials
5.2
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Multiplying 3 or more Polynomials
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Use same technique as you used for numbers:
Multiply any 2 together and simplify the temporary product
Multiply that temporary product times any remaining
polynomial and simplify
-2r(r – 2s)(5r – s)
= -2r(5r2 – 11rs + 2s2)
= -10r3 + 22r2s – 4rs2
5.2
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The Product of Conjugates (Sum and Difference)
(A + B)(A – B) = A2 – B2
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The middle term disappears Always when the binomials
are conjugates (identical, except for middle sign)
Multiplying these is easier than using FOIL!
(x + 4)(x – 4) = x2 – 42 = x2 – 16
 (5 + 2w)(5 – 2w) = 25 – 4w2
 (3x2 – 7)(3x2 +7) = 9x4 – 49
 (-4x – 10)(-4x + 10) = 16x2 – 100
 (6 + 4y)(6 – 4x) = use the foil method
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36 – 24x + 24y – 16xy
5.2
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Thought provoker:
Are these Conjugates?
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(x + 2y)(3xz – 6yz)
= 3x2z – 6xyz + 6xyz – 12y2z
= 3x2z – 12y2z
Why does the middle term disappear?
Because the 2nd binomial conceals a conjugate!
Both terms contain a common factor, 3z :
(x + 2y)(3xz – 3∙2yz)
The middle term ONLY disappears when binomial
conjugates are involved.
5.2
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Squaring a Binomial – Creates a Perfect-Square Trinomial
(A + B)(A + B) = A2 + 2AB + B2
Square the 1st term
Multiply 1st times 2nd, double it, add it
Square the 2nd term, add it
 y  5 2
 y  10 y  25
2
 2 x  3 y 2

1
2
a  3b
 4 x  12 xy  9 y
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4 2
0 . 3 x  7 y 
5.2
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
2
1
4
2
a  3 ab  9 b
2
4
8
 0 . 09 x  4 . 2 xy  49 y
2
12
2
Squaring a Binomial – Creates a Perfect-Square Trinomial
(A – B)(A – B) = A2 – 2AB + B2
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Differences:
Almost the
same
Square the 1st term
Multiply 1st times 2nd, double it, subtract it
Square the 2nd term, add it
 y  5 2
 y  10 y  25
2
3 x  8 y  2

1
5
a  5b
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 9 x  48 xy  64 y
3 2
2
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0 . 6 x  0 . 2 y 
5.2
1
25
2
2
a  2 ab  25 b
2
3
6
 0 . 36 x  0 . 24 xy  0 . 04 y
2
13
2
Examples - board
( y  1)( 1  y ) 
( x  3 y )( x  3 xy  9 y ) 
3
2
2
(a  2b ) 
2
( 5 y  4  3 x )( 5 y  4  3 x ) 
5.2
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Function Notation
If f(x) = x2 – 4x + 5 find:
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a ) f (a )  4
c) f (a  h)  f (a )
 a  4a  5  4
 (a  h)  4(a  h)  5  a  4a  5
 a  4a  9
 a  2 ah  h  4 a  4 h  5  a  4 a  5
2
2
2
2
2
2
2
 2 ah  h  4 h
2
b ) f ( a  3)
 ( a  3)  4 ( a  3)  5
2
 a  6 a  9  4 a  12  5
2
 a  2a  2
2
5.2
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Next …
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Section 5.3 Intro to Factoring
Common Factors, Factoring by Grouping
5.2
16

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