Transcript Unit C - Determining Factors and Roots

C. DETERMINING FACTORS AND
PRODUCTS
Math 10: Foundations and Pre-Calculus
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FP10.1
Demonstrate understanding of factors of whole numbers by
determining the:
prime factors
greatest common factor
least common multiple
principal square root
cube root.
FP10.5
Demonstrate understanding of the multiplication and
factoring of polynomial expressions (concretely, pictorially,
and symbolically) including:
multiplying of monomials, binomials, and trinomials
common factors
trinomial factoring
relating multiplication and factoring of polynomials.
KEY TERMS:
Find the definition of
each of the following
terms:
 Prime Factorization
 Greatest Common
Factor
 Least Common
Multiple
 Perfect Cube
 Cube Root
 Radicand
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Radical
 Index
 Factoring by
Decomposition
 Perfect Square
Trinomial
 Difference of Squares
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1. FACTORS AND MULTIPLES OF WHOLE
NUMBERS
FP10.1
 Demonstrate understanding of factors of whole
numbers by determining the:
 prime factors
 greatest common factor
 least common multiple
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1. FACTORS AND MULTIPLES OF WHOLE
NUMBERS
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Two belts are created, one 12 beads long and the
second 40 beads long. How many beads long
must a belt be for it to created using either
pattern?
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Construct Understanding
p.134
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When a factor of a number has exactly 2 divisors,
1 and itself, the factor is a Prime Factor
For example, the factors of 12 are 1,2,3,4,6,12.
The prime factors are 2 and 3.
To determine the prime factorization of 12, write
as a product of its prime factors.
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To avoid confusing the multiplication sign with
variable x, we use a dot to represent the
multiplication operation.
Example
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The 1st 10 prime numbers are:
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2,3,5,7,11,13,17,19,23,29
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Natural numbers greater than 1 that are not
prime are composite
EXAMPLE
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For 2 or more natural numbers, we can
determine their Greatest Common Factor.
EXAMPLE
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To generate multiples of a number, multiply the
number by the natural numbers, that is 1,2,3,4,5,
etc.
For example lets find the multiples of 26.
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For 2 or more natural numbers, we can
determine their Lowest Common Multiple
When producing multiples for each number the
first common one that comes up is the LCM.
EXAMPLE
EXAMPLE
PRACTICE
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Ex. 3.1 (p. 140) #3-20
2. PERFECT SQUARES, CUBES AND THEIR
ROOTS
FP10.1
 Demonstrate understanding of factors of whole
numbers by determining the:
 principal square root
 cube root.
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2. PERFECT SQUARES, CUBES AND THEIR
ROOTS
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So a perfect cube is a number that can be written
as an integer multiplied by itself three times
Example – 8 is a perfect cube because….
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Which number from 1 to 200 represent perfect
squares?
Which represent perfect cubes?
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Recall that a perfect square is a number that can
be written as an integer multiplied by itself
Example – 36 is a perfect square because….
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Any whole number that can be represented as
the area of a square with a whole number side
length is a perfect square
The side length of the square is the square root of
the area of the square.
Example
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Any whole number that can be represented can
be represented as the volume of a cube with a
whole number edge length is a perfect cube
The edge length of the cube is the cube root of the
volume of the cube.
Example
EXAMPLES
PRACTICE
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Ex. 3.2 (p. 14) #1-14, 17
#1-5, 7-18
3. FACTORS IN POLYNOMIALS
FP10.5
 Demonstrate understanding of the multiplication
and factoring of polynomial expressions
(concretely, pictorially, and symbolically)
including:
 multiplying of monomials, binomials, and
trinomials
 common factors
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3. FACTORS IN POLYNOMIALS
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When we write a polynomial as a product of
factors, we factor the polynomial.
The diagrams on the last slide show that there
are 3 ways to factor the expression 4m+12
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Lets compare multiplying and factoring in
arithmetic and algebra
EXAMPLES
EXAMPLE
PRACTICE
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Ex. 3.3 (p. 154) #3-18
#3-5, 9-21
4. POLYNOMIALS OF THE FORM X2+BX+C
FP10.5
 Demonstrate understanding of the multiplication
and factoring of polynomial expressions
(concretely, pictorially, and symbolically)
including:
 multiplying of monomials, binomials, and
trinomials
 common factors
 trinomial factoring
 relating multiplication and factoring of
polynomials.
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4. POLYNOMIALS OF THE FORM X2+BX+C
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Construct Understanding
p. 159
Pattern – coefficient of c2 is the product of the
coefficients in front of c terms
End (lone) coefficient is the product of the two
lone coefficients
The coefficient in front of c term is the sum of the
two lone coefficients
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Strategies – FOIL, Algebra Tiles, Area Model
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When 2 binomials contain only positive terms,
here are 2 strategies that can be used to
determine the product of the binomials.
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Algebra Tiles (c+5)(c+3)
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Area Model
(h+11)(h+5)
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These strategies show that there are 4 terms in
the product
These terms are formed by applying distributive
property and multiplying each term in the first
binomial by each term in the second binomial.
Example (h+11)(h+5)
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The acronym, and method used to remember the
multiplying strategy is FOIL.
F – 1st term in each mult. together
 O – outside term in each mult. together
 I – inside terms in each mult. together
 L – last term in each mult. together
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Then we add like terms together
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When binomials have negative terms it is hard to
use algebra tiles or the area model so we use
FOIL to solve.
EXAMPLE
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Factoring and multiplying are inverses of each
other.
We can use this to factor a trinomial.
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We can factor using another method which we
will look at with an example.
Factor
z2-12z+35
EXAMPLE
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We should always check our factors by expanding
back out. If we get back what we factored our
factors are correct.
Lets check example b.
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The trinomials in ex 2 are written is descending
order, that is, the terms are written in order from
the term with the greatest exponent first to the
term with the least exponent.
When the order of the terms are reversed, the
terms are written in ascending order.
EXAMPLE
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A trinomial that can be written as the product of
two binomial factors may also have a common
factor.
RULE:
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Also look to see if there is a common factor that
can be taken out of all three terms 1st.
Then factors into 2 binomials.
EXAMPLES
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When we compare factors of a trinomial, it is
important to remember that the order in which
we add terms does not matter
x+a=a+x , for any integer a
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Similarly, the order in which we multiply terms
does not matter.
(x+a)(x+b)=(x+b)(x+a)
PRACTICE
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Ex. 3.5 (p. 165) #2-21
#2-3, 8-23
5. POLYNOMIALS OF THE FORM AX2+BX+C
FP10.5
 Demonstrate understanding of the multiplication
and factoring of polynomial expressions
(concretely, pictorially, and symbolically)
including:
 multiplying of monomials, binomials, and
trinomials
 common factors
 trinomial factoring
 relating multiplication and factoring of
polynomials.
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5. POLYNOMIALS OF THE FORM AX2+BX+C
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To multiply two binomials where the coefficients
of the variables are not 1, we can use the same
strategies as before.
EXAMPLE
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When binomials contain negative terms, it can be
difficult to model their product with tiles.
Using FOIL is then the best method!
EXAMPLE
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The best method for factoring a polynomial of the
form ax2+bx+c is by using a method called
decomposition
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Consider the binomial product: (3h+4)(2h+1)
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We can use FOIL
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To factor 6h2+11h+4 by decomposition, we
reverse the steps above.
Notice that the coefficients of the h-terms have
the product 3(8)=24
This is equal to the product of the coefficients of
the h2-term and the constant term: 6(4)=24
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So, to factor 6h2+11h+4, we decompose the hterm and write is as a sum of 2 terms whose
coefficients have a product of 24.
Back to our example.
EXAMPLE
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To check that the factors are correct, multiply
them.
Check example 3b.
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Since this trinomial is the same as the original
trinomial, the factors are correct.
PRACTICE
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Ex. 3.6 (p. 176) #1-3, 5-20
#1-3, 8-23
6. MULTIPLYING POLYNOMIALS
FP10.5
 Demonstrate understanding of the multiplication
and factoring of polynomial expressions
(concretely, pictorially, and symbolically)
including:
 multiplying of monomials, binomials, and
trinomials
 relating multiplication and factoring of
polynomials.
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6. MULTIPLYING POLYNOMIALS
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Construct Understanding
p. 182
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The distributive property can be used to perform
any polynomial multiplication
Each term of one polynomial must be multiplied
by each term in the other polynomial.
EXAMPLE
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One way to check that your product is correct is
to substitute a number in for the variable in both
the product statement and your simplified
answer.
If both expressions are equal, your product in
likely correct.
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Lets check example 1a
EXAMPLES
PRACTICE
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Ex. 3.7 (p. 185) #1-17
#1-3, 6-22
7. SPECIAL POLYNOMIALS
FP10.5
 Demonstrate understanding of the multiplication
and factoring of polynomial expressions
(concretely, pictorially, and symbolically)
including:
 multiplying of monomials, binomials, and
trinomials
 common factors
 trinomial factoring
 relating multiplication and factoring of
polynomials.
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7. SPECIAL POLYNOMIALS
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Construct Understanding
p. 188
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We say that a2+ab+b2 is a perfect square
trinomial because the 1st and 3rd terms are
perfect squares and the middle term is twice the
product of their square roots
Example
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In the factored form a perfect square trinomial is:
We can use these patterns to factor perfect
square trinomials
EXAMPLE
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If the trinomial is not a perfect trinomial square
we have to factor, straight factoring or
decomposition.
EXAMPLE
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Another example of a special polynomial is a
difference of squares
A difference of squares is a binomial of the form
a2 – b2
We can think of it as a trinomial with a middle
term of 0,
That is, write a2 – b2
as a2 + 0ab – b2
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Lets work through x2 – 25
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Short cut to factoring a difference of squares
EXAMPLE
PRACTICE
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Ex. 3.8 (p.194) #2, 4-18
#2, 7-21