Download Multiplication of Polynomials

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
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For You
(8 x y )(5 x y )
4
7
 40 x y
7
3
2
(2 x 2 yz 5 )( 6 x 5 y10 z 2 )
9
7
11
12 x y z
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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?
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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
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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 3  x  4)( 2 x 2  3x  6)
5x
Leave
Margin
Space
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 2 x 2  3x  6
30 x
  10 x
 6 x  24
3
 3x 2  12 x
15 x 4
5
 x4
3
 2x
 8x
3
2
 10 x 5  15 x 4  28 x 3  11x 2  6 x  24
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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
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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
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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.
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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 2  10 y  25
2 x  3 y 2  4 x 2  12 xy  9 y 2

1
2
a  3b
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4 2
 14 a 2  3ab 4  9b8
0.3x  7 y 
2
5.2
 0.09 x 2  4.2 xy  49 y 2
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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 2  10 y  25
3x  8 y 2  9 x 2  48 xy  64 y 2

1
5
a  5b
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3 2

1
25
0.6 x  0.2 y 
2
5.2
a 2  2ab 3  25b 6
 0.36 x 2  0.24 xy  0.04 y 2
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Examples - board
( y 3  1)(1  y ) 
( x  3 y )( x 2  3xy  9 y 2 ) 
(a  2b) 2 
(5 y  4  3x)(5 y  4  3x) 
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Function Notation
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If f(x) = x2 – 4x + 5 find:
a) f (a)  4
c ) f ( a  h)  f ( a )
 a 2  4a  5  4
 (a  h) 2  4(a  h)  5  a 2  4a  5
 a 2  4a  9
 a 2  2ah  h 2  4a  4h  5  a 2  4a  5
b) f (a  3)
 2ah  h 2  4h
 (a  3) 2  4(a  3)  5
 a 2  6a  9  4a  12  5
 a 2  2a  2
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Next …
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Section 5.3 Intro to Factoring
Common Factors, Factoring by Grouping
5.2
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