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?

a) 5x b) c) 4x² +3y 22

3m n

• 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 parentheses2) Divide each term by the GCF3) Write what remains inside the parentheses

• Factor each polynomial• Ex6. 8x – 6• Ex7. 3x² + 12x• Ex8. 4x³ + 24x² - 16x• Ex9.

• Ex10.

8 5 76 8 4x x x 3 2 4 2 36 3 9a b ab a b

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)

• 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 squares2) the middle term is twice the product of the square roots of the 1st and last terms

• 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

• 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 + c1) Find the product of ac2) Find the two factors of ac that have a sum of b3) Rewrite the trinomial using that sum4) Factor by grouping

• You will have to determine which factor goes 1st and which goes 2nd by trial and error

• 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