Transcript Unit 11: Factoring

Unit 11: Factoring
Section 1: Prime Factorization
• To find the prime factorization of a whole
number, it is best to start by making a factor tree
• Write all answers in standard form (ascending
order of prime factors)
• Ex1. Name all of the factors of 24
• Ex2. Name the first 4 multiples of 5
• By the Common Factor Sum Property, if two
numbers have a factor in common, then the factor
will also go into their sum (and their difference)
• Prime numbers are only divisible by one and the
original number
• Composite numbers have more than 2 factors
• Ex3. Write the prime factorization of 630 in
standard form
• You can use prime factorization to quickly multiply
and divide large numbers
• Ex4. Write the prime factorization of 616 · 980
616
• Ex5. Write the prime factorization of
980
• Section of the book to read: 12-1
Section 2: Common Monomial Factoring
• Common monomial factoring is the reverse of the
distributive property
• You will be factoring out the greatest common
factor from each term and showing what it looked
like before the distributive property was applied
• Ex1. List all of the factors of 9x³
3 2
36x
y
• Ex2. Find the greatest common factor of
5
and 8x y
• Page 767 shows you a visual way to factor
• The factorization is not complete until you have
factored out all of the common factors (that is
why it is best to use the greatest common factor)
2 3
4 5
3 4
32
a
b

24
a
b

36
a
b
• Ex3. Factor
3
5
4 5 2
2 3 4
6 2 3
• Ex4. Factor 20 x yz  35x y z  15x y z  40 x y z
3 2
2 3
12
x
y

22
x
y
• Ex5. Factor and simplify
x
24a3b 4  28a 4b3  16a 2b5
a 2b
• Ex6. Factor and simplify
• Section of the book to read: 12-2
Section 3: Factoring x² + bx + c
• In Unit 2 Section 12 we learned how to FOIL
when multiplying 2 binomials
• Ex1. Multiply (x + 4)(x – 9)
• In this section, we will be doing the reverse of
FOIL, determining what two binomials must have
been multiplied to create the given trinomial
• For this section only, 1 will be the only coefficient
used for the x² term, so the first term in each
binomial will be the variable
• The two last terms will add to be b and multiply
to be c
• Watch your signs!!!
• Ex2. Factor x² + 7x + 12
• Ex3. Factor x² – 13x +40
• Ex4. Factor x² + x – 42
• If your factorization results in two identical
binomials, the trinomial is called a perfect square
trinomial
• Some trinomials cannot be factored using
integers, those are said to be prime
• You can factor a binomial into two binomials if it
is a difference of squares (Unit 2 section 13)
• Ex5. Factor y² - 36
• Ex6. Factor g² + 25
• Sometimes you will need to perform common
monomial factoring first, and then factor the
quadratic that remains (ALWAYS test for this
possibility first)
• Ex7. Factor 4 x5  20 x 4  96 x3
• Section of the book to read: 12-3
Section 4: Factoring ax² + bx + c
• Just like section 3, factoring these types of
quadratics will be the reverse of FOIL
• There is an extra degree of difficulty with these,
because now you must make sure that when you
FOIL the two binomials, every term is correct
• You can start by guessing and checking, but your
basic math skills and number sense should help
you find the answer quickly
• First · first = ax² and
last · last = c
•
•
•
•
•
Ex1. Factor 2x² + 13x + 15
Ex2. Factor 3x² + 11x – 4
Ex3. Factor 6x² + 7x + 2
Ex4. Factor 8x² + 2x – 15
You can use factorization to solve quadratic
equations (must be = 0)
• Once you have factored, set each binomial = 0
and solve for the variable (next section)
• Section of the book to read: 12-5
Section 5: Solving Some Quadratic
Equations by Factoring
• As discussed in the last section, sometimes you
can solve quadratic equations by factoring
• Once factored, set each binomial (and monomial
if there was first common monomial factoring)
equal to 0 and solve for the variable
• The degree of the polynomial tells you how many
potential solutions there are
• Ex1. Solve 3w(2w + 1)(5w – 4) = 0
• In Unit 7 Section 6, we learned that the Quadratic
Formula is one way to solve quadratic equations
(as is factoring)
• The Quadratic Formula can always be used,
factoring can only be used at times
• Solve by factoring
• Ex2. 6x² - 16x = 0
Ex3. 12x² + 17x = -6
• Ex4. Solve 12x² + 11x – 15 = 0
• Ex6. 16x³ + 32x² + 12x = 0
• Section of the book to read: 12-4
Section 6: Which Quadratic Expressions
are Factorable?
• Some quadratic expressions are not factorable
using integers (they are prime)
• When using the quadratic formula, we found that
many solutions were irrational (these would not
be factorable)
• Using ax² + bx + c; if a, b, and c are all rational
numbers and the discriminant (b² - 4ac) is a
perfect square, then you can factor the
expression (Discriminant Theorem)
•
•
•
•
Ex1. Is 8 + 4x² + 3x factorable?
Ex2. Solve 3x² + 5x – 2 = 0
Ex3. Solve 7x² - 6x – 9 = 0
Unless otherwise directed, you may solve using
the quadratic formula or by factoring
• Section of the book to read: 12-8