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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
[email protected]
Chabot Mathematics
§5.2 Multiply§5.2 MultiplyPolyNomialsPolyNomials
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Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §5.1 → PolyNomial Functions
Any QUESTIONS About HomeWork• §5.1 → HW-15
5.1 MTH 55
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Bruce Mayer, PE Chabot College Mathematics
Multiply MonomialsMultiply Monomials Recall Monomial is a term that is a
product of constants and/or variables • Examples of monomials: 8, w, 24x3y
To Multiply MonomialsTo find an equivalent expression for the product of two monomials, multiply the coefficients and then multiply the variables using the product rule for exponents
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From From §1.6§1.6 Exponent Properties Exponent Properties1 as an exponent a1 = a
0 as an exponent a0 = 1
Negative exponents
The Product RuleThe Product Rule
The Quotient Rule
The Power Rule (am)n = amn
Raising a product to a power
(ab)n = anbn
Raising a quotient to a power
.n n
n
a a
b b
.m
m nn
aa
a
.m n m na a a
1, ,
n nn mn
n m n
a b a ba
b aa b a
This sum
mary assum
es that no denom
inators are 0 and that 00 is not
considered. For any integers m
and n
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Bruce Mayer, PE Chabot College Mathematics
Example Example Multiply Monomials Multiply Monomials
Multiply: a) (6x)(7x) b) (5a)(−a) c) (−8x6)(3x4)
Solution a) (6x)(7x) = (6 7) (x x) = 42x2
Solution b) (5a)(−a) = (5a)(−1a)
= (5)(−1)(a a) = −5a2
Solution c) (−8x6)(3x4) = (−8 3) (x6 x4)
= −24x6 + 4 = −24x10
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Bruce Mayer, PE Chabot College Mathematics
(Monomial)•(Polynomial)(Monomial)•(Polynomial)
Recall that a polynomial is a monomial or a sum of monomials.• Examples of polynomials:
5w + 8, −3x2 + x + 4, x, 0, 75y6
Product of Monomial & Polynomial• To multiply a monomial and a polynomial,
multiply each term of the polynomial by the monomial.
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Bruce Mayer, PE Chabot College Mathematics
Example Example (mono)•(poly) (mono)•(poly)
Multiply: a) x & x + 7 b) 6x(x2 − 4x + 5) Solution
a) x(x + 7) = x x + x 7
= x2 + 7x
b) 6x(x2 − 4x + 5) = (6x)(x2) − (6x)(4x) + (6x)(5)
= 6x3 − 24x2 + 30x
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Bruce Mayer, PE Chabot College Mathematics
Example Example (mono)•(poly) (mono)•(poly)
Multiply: 5x2(x3 − 4x2 + 3x − 5)
Solution:
5x2(x3 − 4x2 + 3x − 5) =
= 5x5 − 20x4 + 15x3 − 25x2
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Bruce Mayer, PE Chabot College Mathematics
Product of Two PolynomialsProduct of Two Polynomials
To multiply two polynomials, P and Q, select one of the polynomials, say P. Then multiply each term of P by every term of Q and combine like terms.
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Bruce Mayer, PE Chabot College Mathematics
Example Example (poly)•(poly) (poly)•(poly)
Multiply x + 3 and x + 5
Solution (x + 3)(x + 5) = (x + 3)x + (x + 3)5
= x(x + 3) + 5(x + 3)
= x x + x 3 + 5 x + 5 3
= x2 + 3x + 5x + 15
= x2 + 8x + 15
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Bruce Mayer, PE Chabot College Mathematics
Example Example (poly)•(poly) (poly)•(poly)
Multiply 3x − 2 and x − 1
Solution (3x − 2)(x − 1) = (3x − 2)x − (3x − 2)1
= x(3x − 2) – 1(3x − 2)
= x 3x − x 2 − 1 3x − 1(−2) = 3x2 − 2x − 3x + 2
= 3x2 − 5x + 2
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Bruce Mayer, PE Chabot College Mathematics
Example Example (poly)•(poly) (poly)•(poly)
Multiply: (5x3 + x2 + 4x)(x2 + 3x)
Solution: 5x3 + x2 + 4x
x2 + 3x
15x4 + 3x3 + 12x2
5x5 + x4 + 4x3
5x5 + 16x4 + 7x3 + 12x2
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Bruce Mayer, PE Chabot College Mathematics
Example Example (poly)•(poly) (poly)•(poly)
Multiply: (−3x2 − 4)(2x2 − 3x + 1)
Solution
2x2 − 3x + 1
−3x2 − 4
−8x2 + 12x − 4
−6x4 + 9x3 − 3x2
−6x4 + 9x3 − 11x2 + 12x − 4
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Bruce Mayer, PE Chabot College Mathematics
PolyNomial Mult. SummaryPolyNomial Mult. Summary Multiplication of
polynomials is an extension of the distributive property. When you multiply two polynomials you distribute each term of one polynomial to each term of the other polynomial.
We can multiply polynomials in a vertical format like we would multiply two numbers
(x – 3)(x – 2)x_________
+ 6 –2x+ 0–3xx2_________
x2 –5x + 6
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PolyNomial Mult. By FOILPolyNomial Mult. By FOIL FOIL Method
FOIL Example
(x – 3)(x – 2) = x2 – 5x + 6x(x) + x(–2) + (–3)(x) + (–3)(–2) =
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FOIL ExampleFOIL Example
Multiply (x + 4)(x2 + 3)
Solution
F O I L
(x + 4)(x2 + 3) = x3 + 3x + 4x2 + 12
O
I
F L
= x3 + 4x2 + 3x + 12
The terms are rearranged in descending order for the final answer
FOIL applies to ANY set of TWO BiNomials,
Regardless of the BiNomial Degree
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More FOIL ExamplesMore FOIL Examples
Multiply (5t3 + 4t)(2t2 − 1) Solution:
(5t3 + 4t)(2t2 − 1) = 10t5 − 5t3 + 8t3 − 4t
= 10t5 + 3t3 − 4t Multiply (4 − 3x)(8 − 5x3) Solution:
(4 − 3x)(8 − 5x3) = 32 − 20x3 − 24x + 15x4
= 32 − 24x − 20x3 + 15x4
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Bruce Mayer, PE Chabot College Mathematics
Special ProductsSpecial Products
Some pairs of binomials have special products (multiplication results).
When multiplied, these pairs of binomials always follow the same pattern.
By learning to recognize these pairs of binomials, you can use their multiplication patterns to find the product more quickly & easily
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Bruce Mayer, PE Chabot College Mathematics
Difference of Two SquaresDifference of Two Squares
One special pair of binomials is the sum of two numbers times the difference of the same two numbers.
Let’s look at the numbers x and 4. The sum of x and 4 can be written (x + 4). The difference of x and 4 can be written (x − 4). The Product by FOIL:
(x + 4)(x – 4) = x2 – 4x + 4x – 16 = x2 – 16( )
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Bruce Mayer, PE Chabot College Mathematics
Difference of Two SquaresDifference of Two Squares
Some More Examples
(x + 4)(x – 4) = x2 – 4x + 4x – 16 = x2 – 16
(x + 3)(x – 3) = x2 – 3x + 3x – 9 = x2 – 9
(5 – y)(5 + y) = 25 +5y – 5y – y2 = 25 – y2}What do all
of these have
in common?
ALL the Results are Difference of 2-Sqs:Formula → (A + B)(A – B) = A2 – B2
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General Case F.O.I.L.General Case F.O.I.L.
Given the product of generic Linear Binomials (ax+b)·(cx+d) then FOILing:
Can be Combined IF BiNomials are LINEAR
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Geometry of BiNomial MultGeometry of BiNomial Mult
The products oftwo binomials can be shown in terms of geometry; e.g,(x+7)·(x+5) →(Length)·(Width)
355x
7xx2
Width= (x+5)
Length= (x+7)
(Length)·(Width) = Sum of the areas of the four internal rectangles 7 5 x x 2 5 7 35 x x x
2 12 35 x x
x
55
7x
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Example Example Diff of Sqs Diff of Sqs
Multiply (x + 8)(x − 8) Solution: Recognize from Previous
Discussion that this formula Applies(A + B)(A − B) = A2 − B2
So (x + 8)(x − 8) = x2 − 82
= x2 − 64
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Example Example Diff of Sqs Diff of Sqs
Multiply (6 + 5w)(6 − 5w) Solution: Again Diff of 2-Sqs
Applies → (A + B)(A − B) = A2 − B2
In this Case• A 6 & B 5w
So (6 + 5w) (6 − 5w) = 62 − (5w)2
= 36 − 25w2
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Bruce Mayer, PE Chabot College Mathematics
Square of a BiNomialSquare of a BiNomial
The square of a binomial is the square of the first term, plus twice the product of the two terms, plus the square of the last term.
(A + B)2 = A2 + 2AB + B2
(A − B)2 = A2 − 2AB + B2
These are called perfect-square trinomials
222222: BABABABA NOTE
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Example Example Sq of BiNomial Sq of BiNomial
Find: (x + 8)2
Solution: Use (A + B)2 = A2+2AB + B2
(x + 8)2 = x2 + 2x8 + 82
= x2 + 16x + 64
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Example Example Sq of BiNomial Sq of BiNomial
Find: (4x − 3x5)2
Solution: Use (A − B)2 = A2 − 2AB + B2
In this Case• A 4x & B 3x5
(4x − 3x5)2 = (4x)2 − 2 4x 3x5 + (3x5)2
= 16x2 − 24x6 + 9x10
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Summary Summary Binomial Products Binomial Products
Useful Formulas for Several Special Products of Binomials:
For any two numbers A and B, (A − B)2 = A2 − 2AB + B2
For two numbers A and B, (A + B)2 = A2 + 2AB + B2
For any two numbers A and B, (A + B)(A − B) = A2 − B2.
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Multiply Two POLYnomialsMultiply Two POLYnomials
1. Is the multiplication the product of a monomial and a polynomial? If so, multiply each term of the polynomial by the monomial.
2. Is the multiplication the product of two binomials? If so:
a) Is the product of the sum and difference of the same two terms? If so, use pattern(A + B)(A − B) = A2 − B2
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Multiply Two POLYnomialsMultiply Two POLYnomials
2. Is the multiplication the product of Two binomials? If so:
b) Is the product the square of a binomial? If so, use the pattern (A + B)2 = A2 + 2AB + B2, or (A − B)2 = A2 − 2AB + B2
c) c) If neither (a) nor (b) applies, use FOIL
3. Is the multiplication the product of two polynomials other than those above? If so, multiply each term of one by every term of the other (use Vertical form).
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Example Example Multiply PolyNoms Multiply PolyNoms
a) (x + 5)(x − 5) b) (w − 7)(w + 4)
c) (x + 9)(x + 9) d) 3x2(4x2 + x − 2)
e) (p + 2)(p2 + 3p – 2)
SOLUTION
(x + 5)(x − 5) = x2 − 25
(w − 7)(w + 4) = w2 + 4w − 7w − 28
= w2 − 3w − 28
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Example Example Multiply PolyNoms Multiply PolyNoms
SOLUTION
c) (x + 9)(x + 9) = x2 + 18x + 81
d) 3x2(4x2 + x − 2) = 12x4 + 3x3 − 6x2
e) By columns
p2 + 3p − 2 p + 2
2p2 + 6p − 4 p3 + 3p2 − 2p p3 + 5p2 + 4p − 4
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Function NotationFunction Notation
From the viewpoint of functions, if
f(x) = x2 + 6x + 9
and
g(x) = (x + 3)2
Then for any given input x, the outputs f(x) and g(x) above are identical.
We say that f and g represent the same function
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Example Example ff((a a + + hh) ) − − ff((aa))
For functions f described by second degree polynomials, find and simplify notation like f(a + h) and f(a + h) − f(a)
Given f(x) = x2 + 3x + 2, find and simplify f(a+h) and simplify f(a+h) − f(a)
SOLUTION
f (a + h) = (a + h) 2 + 3(a + h) + 2
= a 2 + 2ah + h
2 + 3a + 3h + 2
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Bruce Mayer, PE Chabot College Mathematics
Example Example ff((a a + + hh) ) − f(− f(aa))
Given f(x) = x2 + 3x + 2, find and simplify f(a+h) and simplify f(a+h) − f(a)
SOLUTION
f (a + h) − f (a) =[(a + h)2 + 3( a + h) + 2] − [a2 + 3a + 2]
= a 2 + 2ah + h 2 + 3a + 3h + 2 − a2 − 3a − 2
= 2ah + h2 + 3h
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Multiply PolyNomials as FcnsMultiply PolyNomials as Fcns
Recall from the discussion of the Algebra of Functions The product of two functions, f·g, is found by
(f·g)(x) = [f(x)]·[g(x)]
This can (obviously) be applied to PolyNomial Functions
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Example Example Fcn Multiplication Fcn Multiplication
Given PolyNomial Functions
( ) 3f x x 2( ) 6 8g x x x Then Find: (f·g)(x) and (f·g)(−3) SOLUTION
23 6 8x x x 3 2 26 8 3 18 24x x x x x
(f · g)(x) = f(x) · g(x)
3 23 10 24x x x
(f · g)(−3)
3 23 3 3 10 3 24 27 27 30 24
0
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WhiteBoard WorkWhiteBoard Work
Problems From §5.2 Exercise Set• 30, 54, 82, 98b, 116, 118
PerfectSquareTrinomialByGeometry
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Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
Remember FOIL By
BIG NOSEDiagram
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
[email protected]
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
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Bruce Mayer, PE Chabot College Mathematics
Graph Graph yy = | = |xx||
Make T-tablex y = |x |
-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6
x
y
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
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Bruce Mayer, PE Chabot College Mathematics
x
y
-3
-2
-1
0
1
2
3
4
5
-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls