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ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-1 – FACTORING INTEGERS
FACTOR – a number that can be divided nicely into your given number
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
FACTOR SET – the group of numbers (i.e. natural, whole, integers, etc.)
**List the positive (integer) factors of each of the following numbers:
a) 54  1, 2, 3, 6, 9, 18, 27, 54
b) 20  1, 2, 4, 5, 10, 20
**List all (integer) factors of each of the following numbers:
a) 30  1, 2, 3, 5, 6, 10, 15, 30
b) 12  1, 2, 3, 4, 6, 12
PRIME NUMBER – a number that can only be divided by one and itself
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**The first several prime numbers include:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, …
**By definition, 1 is NOT a prime number.
**Two is the only even prime number.
PRIME FACTORIZATION – the group of prime numbers that can be multiplied
together to get your number
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**There are two methods of finding the prime factorization of a number:
1) FACTOR TREE – Break down each number on each “branch” into
factors until only prime numbers are left.
2) UPSIDE-DOWN DIVISION – Divide your number by a prime number.
Then continue to divide your quotients by primes until only primes are
left
GREATEST COMMON FACTOR (GCF) – the biggest factor that is common to
two or more numbers
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**One way to find the GCF is to list the factors of each number and find the
biggest one on each list.
**The second way to find the GCF is a three-step process:
1) Do the PRIME FACTORIZATION of each number.
2) Circle pairs of numbers that appear in the prime factorizations of all the
numbers.
3) Multiply one of each pair of common numbers together to get the GCF.
ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-2/5-3 – DIVIDING MONOMIALS/POLYNOMIALS
FACTORING OUT THE GCF (Greatest Common Factor)
**To factor a polynomial, take out the GCF of each of the terms.
**The GCF of the terms includes:
1) The GCF of the coefficients (the biggest number that goes into them)
2) The most of each of the variables in common
**Find the GCF of the following terms:
a) 8x2 and 16xy  GCF = 8x (8 divides into 8 and 16 and the most number
of x’s is x and there are no y’s in common)
b) 5x2y3 and 15xy2  GCF = 5xy2 (5 divides into 5 and 15 and the most
number of x’s in common is x and the most number of y’s in common in
y2)
c) 3xyz and 5xy2z  GCF = xyz (Nothing divides nicely into 3 and 5 and
the most of each variable in common is x, y, and z)
**Factor out the GCF of each of the following polynomials:
a) 5x2 + 15xy – 10xy2 = 5x(x + 3y – 2y2)
b) 8a2b3c – 12ab4c = 4ab3c(2a – 3b)
c) 3x2 + 9x2y = 3x2(1 + 3y)
**Notes about factoring out the GCF:
**Factor out the GCF and keep it out in front. Write down what’s left in
parentheses.
**If everything is factored out of a term in the polynomial, you MUST
write down a one.
**Factoring is the opposite of the distributive property. You can check
your work by distributing your answer out.
SIMPLIFYING FRACTIONS
**To reduce a fraction, cancel out the GCF of the numerator and the
denominator:
a)
b)
12
12 4 3
 the GCF is 4   
16
16 4 4
48
48 16 3
 the GCF is 16 
 
80
80 16 5
**The rule also applies to fractions that have variables:
a)
b)
12 xy
12 xy 4 x 3 y
 the GCF is 4 x 


16 x
16 x 4 x 4
48 xyz
48 xyz 12 xy 4 z
 the GCF is 12 xy 


 4z
12 xy
12 xy 12 xy 1
**If the variables have exponents, the rule for dividing exponents is:
“Subtract the exponents.”
**Simplify the following fractions:
a)
12 x 2 y 3
12 x 2 y 3 4 x 2 y 3 y 2
2
 the GCF is 4 x y 


16 x 3 y
16 x 3 y 4 x 2 y 4 x
b)
48x 3
48x 3 16 x 3x 2
 the GCF is 16 x 


80 xy
80 xy 16 x 5 y
**If a fraction has addition or subtraction in the numerator and/or denominator,
there are three approaches to simplifying:
**OPTION #1  Cancel the GCF out of every piece.
a)
15 x 3  10 x 2
3x 2  2 x
 5 x is the GCF 
 3x 2  2 x
5x
1
**OPTION #2  Split the fraction into parts and simplify individually.
b)
15 x 3  10 x 2 15 x 3 10 x 2 3x 2 2 x




 3x 2  2 x
5x
5x
5x
1
1
**OPTION #3  Factor and cancel
c)
15 x 3  10 x 2 5 x  (3x 2  2 x) 3x 2  2 x


 3x 2  2 x
5x
5x
1
**Notes about simplifying fractions:
**If everything cancels out of the numerator, you must write down a one.
**If everything cancels out of the denominator, a one is there but you do
not have to write it down.
**Watch the negative signs. If the fraction is negative overall, the negative
sign can be in the numerator, denominator of in the middle:
2
2
2


5
5
5
ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-4 – FOIL-ING
FOIL-ING – a method used to distribute out polynomials that are being
multiplied together
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**FOIL is an acronym. The letters stand for:
F  First O  Outside I  Inside L  Last
**In order to FOIL, multiply the first terms in each of the parentheses, multiply
the outside terms, multiply the inside terms, and multiply the last terms in
each of the parentheses.
**Be sure to combine like terms and simplify after FOIL-ing if necessary.
**FOIL each of the following problems:
a) (x + 5)(x – 3) = x2 – 3x + 5x – 15 = x2 + 2x – 15
b) (4x – 1)(2x + 3) = 8x2 + 12x –2x – 3 = 8x2 + 10x – 3
c) (x + 6)(x – 6) = x2 + 6x – 6x – 36 = x2 – 36
QUADRATIC POLYNOMIAL – a polynomial with terms whose greatest power
is two
LINCS Table
1) TERM
3) REMINDING WORD
6) EXAMPLE
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
**QUADTRATIC TERM – the term that contains the power of two
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**LINEAR TERM – the term that contains the power of one (no written power)
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**CONSTANT TERM – the term that consists of just a number
LINCS Table
1) TERM
3) REMINDING WORD
6) EXAMPLE
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-7/5-8 – FACTORING QUADRATICS (w/ coefficient of 1)
QUADRATIC POLYNOMIAL – a polynomial with terms whose greatest power
is two
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**When factoring a quadratic, the goal is to break it into two sets of
parentheses that can be multiplied together to get the original expression.
**Factoring is the opposite of FOIL-ing.
**Quadratics should always be written in descending order (highest power to
lowest power):
ax2 + bx + c
**To factor a quadratic (where a = 1 and c is positive), follow these steps:
1) Write down two sets of parentheses. 
(
)(
)
2) Place an x (or your variable) in each of the parentheses. 
(x )(x )
3) If c is positive, the signs in each parenthesis are the same. The sign on
the middle number (b) will tell you whether they are + or –. 
(x + __ )(x + __ ) or (x – __ )(x – __ )
4) The remaining numbers must multiply together to get the last number (c)
and add (or subtract) to get the middle number (b).
**Factor each of the following quadratics:
a) x2 + 8x + 15 = (x + __ )(x + __ ) = (x + 5)(x + 3)
b) x2 – 10x + 24 = (x – __ )(x – __ ) = (x – 6)(x – 4)
c) x2 + 3x + 4 = …cannot be factored
**Quadratics should always be written in descending order (highest power to
lowest power):
ax2 + bx + c
**To factor a quadratic (where a = 1 and c is negative), follow these steps:
1) Write down two sets of parentheses. 
(
)(
)
2) Place an x (or your variable) in each of the parentheses. 
(x )(x )
3) If c is negative, the signs in each parenthesis are different. 
(x + __ )(x – __ ) or (x – __ )(x + __ )
4) The remaining numbers must multiply together to get the last
number (c) and add (or subtract) to get the middle number (b).
**Factor each of the following quadratics:
a) x2 + 2x – 15 = (x + __ )(x – __ ) = (x + 5)(x – 3)
b) x2 – 10x – 24 = (x + __ )(x – __ ) = (x + 2)(x – 12)
c) x2 + 5x – 4 = …cannot be factored
**Quadratics that cannot be factored are considered to be PRIME.
**If a y and a y2 happens to be part of the quadratic, a y is attached to the
numbers in the parentheses.
**Factor each of the following quadratics:
a) x2 + 8xy + 15y2 = (x + 5y)(x + 3y)
b) x2 + 2xy – 15y2 = (x + 5y)(x – 3y)
ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-9 – FACTORING QUADRATICS (w/out coefficient of 1)
QUADRATIC POLYNOMIAL – a polynomial with terms whose greatest power
is two
**When factoring a quadratic, the goal is to break it into two sets of
parentheses that can be multiplied together to get the original expression.
**Factoring is the opposite of FOIL-ing.
**Quadratics should always be written in descending order (highest power to
lowest power):
ax2 + bx + c
**To factor a quadratic (where a  1 and c is positive), follow these steps:
1) Write down two sets of parentheses. 
(
)(
)
2) Place an x (or your variable) in each of the parentheses. 
( __ x )( __ x )
3) If c is positive, the signs in each parenthesis are the same. The sign on
the middle number (b) will tell you whether they are + or –. 
( __ x + __ )( __ x + __ ) or ( __ x – __ )( __ x – __ )
4) The numbers in front of the x’s are factors that multiply together to get
the number in front of the x2 term (a) 
5) The remaining numbers must multiply together to get the last number (c)
and add (or subtract) in combination with the numbers in front of the
x’s to get the middle number (b).
**Factor each of the following quadratics:
a) 2x2 + 7x + 3 = (2x + __ )(1x + __ ) = (2x + 1)(1x + 3) = (2x + 1)(x + 3)
b) 3x2 – 8x + 4 = (3x – __ )(1x – __ ) = (3x – 2)(1x – 2) = (3x – 2)(x – 2)
c) 2x2 + 7x + 4 = …cannot be factored
**Quadratics should always be written in descending order (highest power to
lowest power):
ax2 + bx + c
**To factor a quadratic (where a  1 and c is negative), follow these steps:
1) Write down two sets of parentheses. 
(
)(
)
2) Place an x (or your variable) in each of the parentheses. 
( __ x )( __ x )
3) If c is negative, the signs in each parenthesis are different. 
( __ x + __ )( __ x – __ ) or ( __ x – __ )( __ x + __ )
4) The numbers in front of the x’s are factors that multiply together to get
the number in front of the x2 term (a) 
5) The remaining numbers must multiply together to get the last number (c)
and add (or subtract) in combination with the numbers in front of the
x’s to get the middle number (b).
**Factor each of the following quadratics:
a) 2x2 + x – 15 = (2x + __ )(1x – __ ) = (2x – 5)(1x + 3) = (2x – 5)(x + 3)
b) 5x2 – 7x – 6 = (5x + __ )(1x – __ ) = (5x + 3)(1x – 2) = (5x + 3)(x – 2)
c) 2x2 + 9x – 4 = …cannot be factored
**Quadratics that cannot be factored are considered to be PRIME.
**If a y and a y2 happens to be part of the quadratic, a y is attached to the
numbers in the parentheses.
**Factor each of the following quadratics:
a) 2x2 + 7xy + 3y2 = (2x + y)(x + 3y)
b) 5x2 – 7xy – 6y2 = (5x + 3y)(x – 2y)
ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-5 – DIFFERENCE OF TWO SQUARES
SQUARE ROOT – the square root of a number (or expression) is the number (or
expression) that can be multiplied by itself to get the given number (or
expression)
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**Find the square root of the following numbers:
a) 64 = 8 …because 8  8 = 64
b) 144 = 12 …because 12  12 = 144
c) 10 = a weird decimal and is not a PERFECT SQUARE
**PERFECT SQUARE – a number (or expression) that has a nice square root;
in other words the square root is not a weird decimal
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
3) REMINDING WORD
6) EXAMPLE
**To find the square root of an expression:
1) Find the square root of the coefficient.
2) Take half of the exponent for each of the variables.
**Find the square root of the following expressions:
a) 9x2 = 3x …because 3x3x = 9x2
b) 36x6y4 = 6x3y2 …because 6x3y26x3y2 = 36x6y4
2) DEFINITION
**Notes about square roots:
**You cannot take the square root of a negative number.
**Relatively few numbers are perfect squares.
**Numbers that have decimal square roots are okay, however the decimal is
not the exact square root.
“DIFFERENCE of TWO SQUARES” is a special case of factoring quadratics.
**In this situation, the two sets of parentheses are almost identical except for the
fact that the signs in each set of parentheses are opposite.
**One approach is to factor this situation like a normal quadratic.
**A second approach is to identify the original problem as a Difference of Two
Squares by looking at the terms for several clues:
1) There must be only two terms in the polynomial.
2) The two terms must be subtracted (difference).
3) The first and last terms must have nice square roots.
**Factor each of the following quadratics:
a) x2 – 36 = (x – 6)(x + 6)
b) 4x2 – 9 = (2x – 3)(2x + 3)
c) 9x2 – 16y2 = (3x – 4y)(3x + 4y)
ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-6 – SQUARING BINOMIALS
SQUARE ROOT – the square root of a number (or expression) is the number (or
expression) that can be multiplied by itself to get the given number (or
expression)
**To find the square root of an expression:
1) Find the square root of the coefficient.
2) Take half of the exponent for each of the variables.
**Find the square root of the following expressions:
a) 16x8 = 4x4 …because 4x44x4 = 16x8
b) 49x2y10 = 7xy5 …because 7xy57xy5 = 49x2y10
**PERFECT SQUARE – a number (or expression) that has a nice square
root; in other words the square root is not a weird decimal
**Notes about square roots:
**You cannot take the square root of a negative number.
**Relatively few numbers are perfect squares.
**Numbers that have decimal square roots are okay, however the
decimal is not the exact square root.
“SQUARING BINOMIALS” is a special case of factoring quadratics.
**In this situation, the two sets of parentheses happen to be identical.
**One approach is to factor this situation like a normal quadratic. Be aware that
if the parentheses are identical, it is this situation.
**A second approach is to look at the terms and look for several clues:
1) There must be three terms in the polynomial that are in descending order.
2) The last term (constant) must be positive.
3) The first and last terms must have nice square roots.
4) The middle term must be equal to two times the square roots of the
first and last terms.
**Factor each of the following quadratics:
a) x2 + 10x + 25 = (x + 5)(x + 5)
b) 4x2 – 12x + 9 = (2x – 3)(2x – 3)
c) 9x2 – 24xy + 16y2 = (3x – 4y)(3x – 4y)
ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-10 – FACTORING BY GROUPING
“FACTORING BY GROUPING” consists of using the distributive property and
factoring out the GCF to factor an expression
**Follow these steps to factor by grouping:
1) Group the terms into two pairs of terms where each pair has something in
common.
2) Factor out the GCF for each pair and write down what is left in
parentheses. The two sets of parentheses should be the same.
3) Factor out the group (the parentheses) from each pair.
4) You should be left with two sets of parentheses.
**Factor each of the following expressions by grouping:
a) 5x + wx + wy + 5y =
5x + 5y + wx + wy =
5(x + y) + w(x + y) = (x + y)(5 + w)
b) 2ab – 6ac + 3b – 9c =
2ab + 3b – 6ac – 9c =
b(2a + 3) – 3c(2a + 3) = (2a + 3)(b – 3c)
ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-11 – SEVERAL METHODS OF FACTORING
FACTORING COMPLETELY – a polynomial that is factored as much as
possible and consists of a monomial and prime polynomials (or any combination
of those)
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**Follow these steps to factor an expression completely:
1) Look to factor out the GCF first.
2) Try and identify the expression as a “Difference of Two Squares” or a
“Squaring Binomial”. If this is the case, factor.
3) Next, look to factor the expression as a quadratic.
4) If the expression has four or more terms, try factoring by grouping.
5) After factoring, make sure each of the terms is prime (cannot be factored
any further).
6) Multiply out the terms to see if it equals the original expression.
**Factor the following expressions completely:
a)
b)
EXPRESSION
TYPE OF FACTORING
a2bc – 4bc + a2b – 4b
b[a2c – 4c + a2 – 4]
b[c(a2 – 4) + 1(a2 – 4)]
b[(a2 – 4)(c + 1)]
b(a – 2)(a + 2)(c + 1)
Difference of 2 Squares
EXPRESSION
TYPE OF FACTORING
5a2 – 20b2
5(a2 – 4b2)
5(a – 2b)(a + 2b)
Factor out GCF
Difference of 2 Squares
Factor out GCF
Factor by Grouping
c)
EXPRESSION
32b4 – 48b3c + 18b2c2
2b2(16b2 – 24bc + 9c2)
2b2(4b – 3c)(4b – 3c)
2b2(4b – 3c)2
TYPE OF FACTORING
Factor out GCF
Squaring Binomial
ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-12 – SOLVING QUADRATIC EQUATIONS
POLYNOMIAL EQUATION – an equation whose sides consist of polynomials
**LINEAR EQUATION – an equation whose highest degree is one; the
form of a linear equation is ax + b = 0
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**You solved linear equations in Chapter 3.
**QUADRATIC EQUATION – an equation whose highest degree is two; the
form of a quadratic equation is ax2 + bx + c = 0
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**CUBIC EQUATION – an equation whose highest degree is three; the form of
a cubic equation is ax3 + bx2 + cx + d = 0
LINCS Table
1) TERM
3) REMINDING WORD
6) EXAMPLE
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
**In order to solve equations that have exponents (quadratic, cubic, etc.), you
must factor.
**Solving equations by factoring is based on the zero-product property.
**ZERO-PRODUCT PROPERTY – this property says that when two (or
more) factors are multiplied together, if any one of the factors is zero,
the answer is zero
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**To solve a quadratic (or higher powered) equation, you must:
1) Move all pieces to one side of the equal sign and set it equal to zero.
2) Write the terms in descending order.
3) Factor the expression completely.
4) Set each factored piece equal to zero and solve the mini-equation.
**Notes about solving equations:
**The highest power will tell you the number of answers (i.e. a quadratic
has an exponent of 2, therefore there will be 2 answers).
**Occasionally you may have answers that are repeated. You only have to
write it down once, but both count toward the total number of
answers.
**If you factor out a GCF, this term will give you an answer of 0.
**The equation must be set equal to zero in order for this technique to
work.
**ROOT – another word that means the answer to an equation
LINCS Table
1) TERM
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
3) REMINDING WORD
6) EXAMPLE
**MULTIPLE ROOT – an answer that is repeated
LINCS Table
1) TERM
3) REMINDING WORD
6) EXAMPLE
4) LINKING STORY
5) LINKING PICTURE
2) DEFINITION
ALGEBRA I NOTES: CHAPTER FIVE
SECTION 5-13 – WORD PROBLEMS (w/ FACTORING)
NO NOTES
ALGEBRA I NOTES: CHAPTER FIVE
SUMMARY
Factor –
Factor Set –
Prime Number –
Prime Factorization –
Factor Tree –
Upside-Down Division –
GCF (Greatest Common Factor) –
Factoring out GCF –
Distributing –
FOIL-ing –
Quadratic Polynomial –
Quadratic Term –
Linear Term –
Constant –
Factoring Quadratics
Square Root –
Perfect Square –
Difference of Two Squares –
Squaring Binomials –
Factoring by Grouping –
Factoring Completely (multi-step factoring) –
Polynomial Equation –
Linear Equation –
Quadratic Equation –
Cubic Equation –
Zero-Product Property –
Root –
Multiple Root –