Ch. 9 notes

1 Algebra I Chapter 9 Notes Mrs. Cataldo CHAPTER 9 NOTES SHORT QUIZ: 1. Write in exponential form using a negative exponent. a. 3/x b. 1/100 c. y/x² 2. Simplify a. ( y – 4) – 6 b. 4(x – 3) c. y + x² ‐x + y – 2x² d. (‐3y³)² 3. Find the value of the expression when x = ‐1 and y = 4. ‐5(2x ) 9.1 Monomials So far we have seen equations like 5x + 2y = 53. Each part of a number sentence like this one separated by addition is a ___________. This equation has ___ terms on the left of the equals sign and ____ terms on the right. Each term has a variable to some power and each term has a coefficient. The coefficient is the number of times the variable is taken to that power. If we have 6 apples and 5 oranges in a basket we could represent apples as “a” and oranges as “o”. We could write 6a + 5o as an expression of the fruit in our basket. The coefficient of the apples is 6, the number of apples we have. In the above expression, the coefficient of x is ___ and the coefficient of y is ___. Another aspect of a term is the degree of the term. That is the ________ to which the variable is raised. eg. For 6x³, the coefficient of this term is ____ and the degree is _____. Let’s look at the first equation again. The degree of the term 5x is _____. 2 The degree of the 2y term is ____. What about 53, can we write that with a variable? What is the value of ? ___ So we can write 53 as 53x , a term with the degree of ____, and the coefficient of ___. What about 0? We can write 0 in so many ways that we don’t consider 0 to have a specific degree. 0x = 0 0x = 0 etc.. An equation or expression with one term is called a MONOMIAL. eg. 16x² is a monomial A _______________________ in one variable (as above) is an expression of the form ax where a is the coefficient of the monomial and n is the degree of the monomial. NOTE: n must be a positive integer. Which of these are monomials? 7x² (½)x x‐2 ‐2x 2/x 2x As you may remember, terms that are alike can be added together. If they are not like terms they cannot be added. eg. 4x + 5x can be written as ______ 4x² + 5x cannot be written any other way In order for terms to be considered like terms, they must not only have the same variable but also the same degree. Simplify if possible: 2x³ ‐2x³ 2x6 + x² Terms (aka monomials) can be multiplied whether they are like or not. 3 Simplify if possible; 4x³· x³ (3y)(xy²) (4x5)² 4 9.2 Polynomials Monomial = “one name” so each term can be considered a monomial. Polynomial = “many names” so an expression with one or more terms is a polynomial. A _____________________ is either a monomial or an expression indicating the addition and/or subtraction of two or more monomials. The monomials are called ____________ of the polynomial. eg. 2y³ is a monomial and a polynomial 2y³ + x + 1 is a polynomial with three terms Which are polynomials? 5x² + 10x € 10x + 5 ‐8x € 10x + 5 From now on we will write our polynomials as the addition of various terms. Write € 2v6 − v3 − 3 as a sum of monomials. __________________________ Each term in a polynomial has a degree. By convention we write a polynomial in the order of highest to lowest degree. eg. y + y³ ‐5y² ‐1 is written y³ + ‐5y² + y + ‐1 The degree of the polynomial is the degree of the highest degree term. What is the degree of the polynomial above? _____ Before determining the degree and number of terms in a polynomial, it should be simplified. eg. Simplify and write as a polynomial using the convention mentioned above. How many terms are there? What is the degree of the polynomial? 4x(2x ‐6x ) + x ‐3x +1 The value of a polynomial is determined by substituting a given value for the variable in the expression and evaluating it. What is the value of the above polynomial if x = ‐1? 5 9.3 Adding and Subtracting Polynomials When we add or subtract polynomials, we add or subtract like terms ONLY. It is easiest to keep track by lining up like terms vertically in order of descending degree. eg. Add 4x² + x + 5 and x² ‐7x + 3 First write as addition expressions, then line up vertically with like terms below each other. 4x² + x + 5 + x² + ‐7x + 3 5x² + ‐6x + 8 Add x³ ‐7x² ‐1 and 3x² +8x. Subtract 7x² ‐9 from 10x² + x + 2. 6 9.4 Multiplying Polynomials Multiply 53 Write in expanded notation: 50 + 3 x 27 x 20 + 7 The second way we multiplied looks like a two term polynomial. We multiply each two terms and add the four products. This can be illustrated with a square. 50 3 20 7 Multiply (x + 4)(x + 3) x 4 x 3 Multiply (5x + 1)(2x – 7) 5x 1 2x ‐7 7 Go over Set 2 problems 4, 5a, 6a, 7a, 8a 8 Short Quiz: Match the like terms. 9 4x x² xy 9x 3x³ 5xy 22 1.5x³ (½)x² Multiply (x + 2)(4 – x) using a square and vertically. Multiply (x²)(x + 2xy – y). 9 9.5 Multiplying polynomials horizontally Multiply (2x + 6)(x + 5). Square method: Vertically: 2x 6 x 5 New horizontal method. Multiply each term of one polynomial by each _______ of the other. Add _______ terms. (2x + 6)(x+5) = (2x)(x) + (2x)(5) + (6)(x) + (6)(5) = 2x² + 10x + 6x + 30 = 2x² + 16x + 30 Practice. Multiply the following. (x² ‐7)(6x – 1) 10 (x² ‐5x + 25)(x + 5) (2x + 7)(x² ‐3x + 2) 9.6 Squaring Binomials If a monomial has one term and a polynomial had many terms, how many terms does a binomial have? ____ How many terms does a trinomial have? _____ Multiply (x + y)(x + y). Any time you square a binomial with added terms, you will get this form. You will need to memorize this. If you substitute a number for y, eg y = 7, what do you get? Multiply (x +7)(x +7). Now change and multiply (x³ + 2x)(x³ + 2x). Notice the pattern. Multiply (x ‐y)(x – y). 11 Any time you square a binomial with subtracted terms, you will get this form. You will need to memorize this. How is the result different from the one you get when squaring a binomial with added terms?__________________ Again, if you substitute a number for y, eg y = 2, what do you get? Multiply (x – 2)(x – 2). Or change x to 3x and y to 5. Multiply (3x – 5)(3x – 5). The last case is not a square of a binomial but a special case where you multiply the added terms by the subtracted terms. This pattern should also be memorized. Multiply (x + y)(x – y) What happened to the middle term? __________________ Find the product of (4x +9) and (4x – 9). Homework. Question 5 asks you to make a diagram to illustrate the products. A diagram is a square. Do 5a (x + 9)². 12 Answer questions 6. Do 7 a, b, d together. Do 8a. Do 10a. 13 9.7 Dividing Polynomials Go over example of the reverse of x 2 3x 3x² 6x 5 5x 10 14 Examples: p. 435 5a, 6a, 9a