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

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 + 24Ex4. 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

Factor each polynomial Ex6. 8x – 6 Ex7. 3x² + 12x Ex8. 4x³ + 24x² - 16x Ex9. Ex10.

Section 9 – 3 Multiplying Binomials To multiply polynomials, you must multiply EVERY term in the 1 st polynomial by EVERY term in the 2 nd 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 2 nd 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 2 nd terms in the binomials are negative If the c value is negative, then one binomial has a positive 2 nd term and the other has a negative 2 nd term (to be determined by the value of b) Factor each trinomial into two binomials. Ex2. m² - 9m + 20Ex3. x² + 13x – 48 Ex4. d² + 17dg – 60g²Ex5. 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 1 st and last terms are perfect squares 2) the middle term is twice the product of the square roots of the 1 st 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 1 st two terms have a factor in common and the 2 nd 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 + 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 1 st and which goes 2 nd 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