Download Chapter 9 Polynomials and Factoring

Chapter 9 Polynomials and Factoring
•A monomial is an expression that contains
numbers and/or variables joined by multiplication
(no addition or subtraction at all)
•The degree of the monomial is found by adding all
of the exponents of all of the variables (not the
exponents of the numerical terms)
•Ex1. Is each expression a monomial? If so, what is
its degree?
2 2
a) 5x
b)  m n
c) 4x² +3y
3
• If an expression has two terms (two monomials)
it is a binomial
• If an expression has three terms (three
monomials) it is a trinomial
• A polynomial is any monomial or sum or
difference of two or more monomials
• Binomials and trinomials are types of
polynomials
• Standard form of a polynomial is when you write
the polynomial with the degrees of each term in
descending order
• To find the degree of a polynomial, find the
degree of each term and the highest degree is
the degree for the entire polynomial
• Study the green chart on page 457
• You can only add and subtract like terms (the
variables and their exponents match exactly)
• Ex2. (3x² + 5x – 8) + (6x² + x + 11)
• Ex3. (5x² + 3x – 4) – (2x² + 2x – 8)
• Ex4. (3x³ + 2x² + 5) – (7x³ – 5x + 2)
Section 9 – 2 Multiplying and Factoring
• When multiplying a monomial by any polynomial,
distribute the monomial to every term of the
polynomial and simplify if you can
• Ex1. 5x²(3x² + 8x – 9)
• Ex2. -3w(w² + 8w + 5)
• To find the greatest common factor (GCF) of
monomials, identify the largest number and
variable(s) that divide evenly into every term and
multiply them together for the GCF
• To find the GCF, you may have to write out the
prime factorization of each term
• Find the GCF of the terms of each polynomial
• Ex3. 9x + 24
Ex4. 5a² + 20a
• Ex5. 6x³ + 4x² - 8x
• To factor a polynomial:
1) Find the GCF of the terms and write that on
the outside of a set of parentheses
2) Divide each term by the GCF
3) Write what remains inside the parentheses
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Factor each polynomial
Ex6. 8x – 6
Ex7. 3x² + 12x
Ex8. 4x³ + 24x² - 16x
Ex9. 6 x8  8 x 5  4 x 7
• Ex10. 6a b  3ab  9a b
3 2
4
2 3
Section 9 – 3 Multiplying Binomials
• To multiply polynomials, you must multiply EVERY
term in the 1st polynomial by EVERY term in the
2nd polynomial and then simplify
• A mnemonic device to help multiply two
binomials is FOIL (First Outer Inner Last)
• Multiply. Use FOIL and then simplify.
• Ex1. (x + 4)(x + 3)
• Ex2. (x – 5)(x + 8)
• Ex3. (2x + 1)(x – 7)
• When multiplying polynomials, you should write
your answers in standard form
• Distribute. Simplify each product. Write in
standard form.
• Ex4. (x + 6)(2x² + 3x – 5)
• Ex5. (3x – 5)(4x² – x + 8)
• Ex6. (4m + 2)(3m² + 5m – 6)
Section 9 – 4 Multiplying Special Cases
• Shortcut to finding the square of a binomial:
(a + b)² = a² + 2ab + b²
and
(a – b)² = a² – 2ab + b²
• You don’t have to take the time to FOIL it out, you
can just use the shortcut (in these cases)
• Ex1. (x + 5)²
• Ex2. (3m – 2)²
• Ex3. (2x – 3y)²
• You can use this to find the square of a whole
number mentally
• Ex4. Find 62² mentally
• The difference of squares: a² – b² = (a + b)(a – b)
• Notice that the outer and inner terms cancel out
• Multiply and simplify
• Ex5. (x + 4)(x – 4)
• Ex6. (m – 7)(m + 7)
• Ex7. (3x² + y²)(3x² – y²)
Section 9 – 5 Factoring Trinomials of the
Type x² + bx + c
• Factoring a trinomial of the type x² + bx + c is the
reverse of FOIL (I call it LIOF)
• You must determine what two binomials will
multiply together to make that trinomial
• The two 2nd terms must multiply to equal c and
add to equal b (from x² + bx + c)
• Ex1. Factor x² + 6x + 8
• All numbers in the question are positive, so all
numbers in the answer are positive for Ex1.
• If the c value is positive and b is positive, then all
numbers in the binomials are positive
• If the c value is positive and b is negative, then the
2nd terms in the binomials are negative
• If the c value is negative, then one binomial has a
positive 2nd term and the other has a negative 2nd
term (to be determined by the value of b)
• Factor each trinomial into two binomials.
• Ex2. m² - 9m + 20
Ex3. x² + 13x – 48
• Ex4. d² + 17dg – 60g²
Ex5. n² + 6n – 27
Section 9 – 6 Factoring Trinomials of the
Type ax² + bx + c
• When you factor a trinomial with a coefficient
with the x² term, you follow similar steps as in the
last section, but you have more things to consider
• You must make sure that when you FOIL out the
binomials they make the given trinomial
• Be sure to test to make sure everything works by
FOILing it out before you move on to the next one
• Open your book to page 486 (example 1)
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Factor
Ex1. 20x² + 17x + 3
Ex2. 3n² - 7n – 6
Ex3. 6x² + 11x – 10
Ex4. 2y² - 5y + 2
Section 9 – 7 Factoring Special Cases
• a² + 2ab + b² and a² – 2ab + b² are perfect
square trinomials (these are each a binomial
squared)
• Ex1. Factor x² + 12x + 36
• Ex2. Factor x² – 14x + 49
• How to identify a perfect square trinomial
1) the 1st and last terms are perfect squares
2) the middle term is twice the product of the
square roots of the 1st and last terms
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Ex3. Factor 16h² + 40h + 25
Ex4. Factor 36x² + 84x + 49
Remember that (a – b)(a + b) = a² – b²
So, if you have a question that is the difference of
two squares, you can factor it into two binomials
• Both terms must be perfect squares and they
must be separated by subtraction
• Ex5. Factor 4x² – 9
• Ex6. Factor 64m² – 25n²
• Sometimes you will have to factor out a term
before you do any further factoring (always check
for this first)
• Ex7. Factor 5x² – 80
• Ex8. Factor 3x² + 24x + 48
Section 9 – 8 Factoring by Grouping
• You can use the distributive property to factor by
grouping if two groups of terms have the same
factor
• For instance, if you have a polynomial with 4
terms and the 1st two terms have a factor in
common and the 2nd two terms have a factor in
common, you can factor in two groups of two
• The goal is that what remains will be identical in
both sets, which allows you to factor one more
time
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Factor
Ex1. 4x² + 8x + 5x + 10
Ex2. 3m² – 15m + 7m – 35
Just like before, if you can factor out a single term
from all terms before you begin, you should do
that first and then see how to factor further
• Ex3. Factor 4x² – 24x + 10x – 60
• You may have to use factoring by grouping to
factor a trinomial
• You can separate the middle term of a trinomial
to two terms that add to be the middle term (the
two terms that would result when you FOILed
before you simplified)
• Use ax² + bx + c
1) Find the product of ac
2) Find the two factors of ac that have a sum of b
3) Rewrite the trinomial using that sum
4) Factor by grouping
• You will have to determine which factor goes 1st
and which goes 2nd by trial and error
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Factor by grouping
Ex4. 24x² + 25x – 25
Ex5. 4y² + 33y – 70
Before you begin your homework, read the box
outlined in orange on the bottom of page 498