the product of two binomials can be found by multiplying each term in one binomial by each term in...
TRANSCRIPT
Expanding Polynomials And
Common Factoring
Review
Expanding Polynomials• The product of two binomials can be found by multiplying EACH term in one binomial by EACH term in the other binomial• Then, simplify (collect like terms)
A B C D
Angelina and Brad go to the movies, where they meet Courtney and David.
If they were to all shake hands with the people they are just meeting…
who would shake hands with who?
A B C D
A B C D
A and C
A and D
B and C
B and D
Expanding polynomials works the same way!
Example 1: Expand and simplify.
a)
b)
3(x 2)
2y(y 1)
3x 6
In this case, the 2y is multiplied by y and the 2y is multiplied by 1.2y2 2y
In this case, the 3 ‘meets’ the x and the 3 ‘meets’ the 2.
c) (x 1)(x 4)
(2x 4)(3x 7y 8)d)
e) (x 4)(2x 3)(5x 1)
Common Factoring•When factoring polynomial expressions, look at both the numerical coefficients and the variables to find the greatest common factor (G.C.F.)• Look for the greatest common numerical factor and the variable with the highest degree of the variable common to each term•To check that you have factored correctly, EXPAND your answer (because EXPANDING is the opposite of FACTORING!)
Example 2: Factor.a)
b)
c)
2x2 8x
9x2y 3xy2
12m3n2 6m4n3 4m2n5 2m2n2
Exponent Laws
Terminology• In multiplication questions, the terms that are multiplied together are called factors
Example: 12 = 6 x 2 6 and 2 are factors of 1212 = 4 x 3 4 and 3 are also factors of 12
• A repeated multiplication of equal factors (the same number) can b expressed as a power
Example: 3 x 3 x 3 x 3 = 34 34 is the power 3 is the base 4 is the exponent
Examples
63 = 6 x 6 x 6 52 + 32 = (5 x 5) + (3 x 3) = 36 x 6 = 25 + 9 = 216 = 34
The Power of Negative Numbers
• There is a difference between –32 and (–3)2
• The exponent affects ONLY the number it touches
So, –32 = –(3 x 3), but (–3)2 = (–3) x (–3) = –9 = 9
Exponent Laws:
•When you multiply powers with the SAME base, you add the exponents
Ex. 22 + 23 = 2 x 2 x 2 x 2 x 2= 25
= 32
Product Rule:
Exponent Laws:
•When you divide powers with the SAME base, you subtract the exponents
Ex. 25 22 = (2 x 2 x 2 x 2 x 2) (2 x 2)= 8= 23
Quotient Rule:
Exponent Laws:
•Any exponent raised to the exponent zero is equal to one
Ex. 20 = 1 990 = 1 1234560 = 1
…. Why is this?Think, pair, share.
Turn to your partners and brainstorm about this for 2 minutes.
Hint: Think about the quotient rule…
Exponent Laws:
By applying the quotient rule….
But we also know that any number divided by itself is 1 1
Exponent Laws:
•When finding the power of a product, apply the exponent to each number in the product
Ex. (2 x 3)2 = 22 x 32
= 4 x 9= 36
Rule:
*Does this rule apply to addition/subtraction?*
Exponent Laws:
Ex. (3 + 2)3 33 + 23
= (5)3 = 9 + 8 = (125) = 17
No, this does NOT work with addition/subtraction
But
Exponent Laws:
•When finding the power of a fraction, apply the exponent to each number in the fraction
Ex. = = = 4
Rule:
Examples: Simplify.
a) b)
c) d)
