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Transcript of [email protected] MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College...
![Page 1: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/1.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§5.2 Multiply§5.2 MultiplyPolyNomialsPolyNomials
![Page 2: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/2.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt2
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
![Page 3: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/3.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt3
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
![Page 4: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/4.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt4
Bruce Mayer, PE Chabot College Mathematics
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
![Page 5: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/5.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt5
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
![Page 6: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/6.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt6
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.
![Page 7: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/7.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt7
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
![Page 8: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/8.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt8
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
![Page 9: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/9.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt9
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.
![Page 10: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/10.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt10
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
![Page 11: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/11.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt11
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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[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt12
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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[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt13
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
![Page 14: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/14.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt14
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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[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt15
Bruce Mayer, PE Chabot College Mathematics
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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[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt16
Bruce Mayer, PE Chabot College Mathematics
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
![Page 17: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/17.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt17
Bruce Mayer, PE Chabot College Mathematics
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
![Page 18: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/18.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt18
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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[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt19
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( )
![Page 20: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/20.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt20
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
![Page 21: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/21.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt21
Bruce Mayer, PE Chabot College Mathematics
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
![Page 22: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/22.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt22
Bruce Mayer, PE Chabot College Mathematics
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
![Page 23: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/23.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt23
Bruce Mayer, PE Chabot College Mathematics
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
![Page 24: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/24.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt24
Bruce Mayer, PE Chabot College Mathematics
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
![Page 25: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/25.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt25
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
![Page 26: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/26.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt26
Bruce Mayer, PE Chabot College Mathematics
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
![Page 27: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/27.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt27
Bruce Mayer, PE Chabot College Mathematics
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
![Page 28: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/28.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt28
Bruce Mayer, PE Chabot College Mathematics
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.
![Page 29: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/29.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt29
Bruce Mayer, PE Chabot College Mathematics
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
![Page 30: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/30.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt30
Bruce Mayer, PE Chabot College Mathematics
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).
![Page 31: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/31.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt31
Bruce Mayer, PE Chabot College Mathematics
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
![Page 32: BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.](https://reader030.fdocuments.us/reader030/viewer/2022032708/56649e615503460f94b5d315/html5/thumbnails/32.jpg)
[email protected] • MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt32
Bruce Mayer, PE Chabot College Mathematics
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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Bruce Mayer, PE Chabot College Mathematics
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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Bruce Mayer, PE Chabot College Mathematics
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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Bruce Mayer, PE Chabot College Mathematics
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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Bruce Mayer, PE Chabot College Mathematics
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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Bruce Mayer, PE Chabot College Mathematics
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, PELicensed Electrical & Mechanical Engineer
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