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Polynomials and Factoring The basic building blocks of algebraic expressions.

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1 Polynomials and Factoring The basic building blocks of algebraic expressions

2 The height in feet of a fireworks launched straight up into the air from (s) feet off the ground at velocity (v) after (t) seconds is given by the equation: -16t 2 + vt + s Find the height of a firework launched from a 10 ft platform at 200 ft/s after 5 seconds. -16t 2 + vt + s -16(5) (5) + 10 = feet

3 In regular math books, this is called “substituting” or “evaluating”… We are given the algebraic expression below and asked to evaluate it. x 2 – 4x + 1 We need to find what this equals when we put a number in for x.. Like x = 3 Everywhere you see an x … stick in a 3 ! x 2 – 4x + 1 = (3) 2 – 4(3) + 1 = 9 – = -2

4 You try a couple Use the same expression but let x = 2 and x = -1 What about x = -5 ? Be careful with the negative! Use ( )! x 2 – 4x + 1 = (-5) 2 – 4(-5) + 1 = 46

5 That critter in the last slide is a polynomial. x 2 – 4x + 1 Here are some others x 2 + 7x – 3 4a 3 + 7a 2 + a nm 2 – m 3x – 2 5

6 For now (and, probably, forever) you can just think of a polynomial as a bunch to terms being added or subtracted. The terms are just products of numbers and letters with exponents. As you’ll see later on, polynomials have cool graphs.

7 Some math words to know! monomial – is an expression that is a number, a variable, or a product of a number and one or more variables. Consequently, a monomial has no variable in its denominator. It has one term. (mono implies one). 13, 3x, -57, x 2, 4y 2, -2xy, or 520x 2 y 2 (notice: no negative exponents, no fractional exponents) binomial – is the sum of two monomials. It has two unlike terms (bi implies two). 3x + 1, x 2 – 4x, 2x + y, or y – y 2

8 trinomial – is the sum of three monomials. It has three unlike terms. (tri implies three). x 2 + 2x + 1, 3x 2 – 4x + 10, 2x + 3y + 2 polynomial – is a monomial or the sum (+) or difference (-) of one or more terms. (poly implies many). x 2 + 2x, 3x 3 + x 2 + 5x + 6, 4x + 6y + 8 Polynomials are in simplest form when they contain no like terms. x 2 + 2x x 2 – 4x when simplified becomes 4x 2 – 2x + 1 Polynomials are generally written in descending order. Descending: 4x 2 – 2x + 1 (exponents of variables decrease from left to right) The ending of these words “nomial” is Greek for “part”. Constants like 12 are monomials since they can be written as 12x 0 = 12 · 1 = 12 where the variable is x 0.

9 The degree of a monomial - is the sum of the exponents of its variables. For a nonzero constant, the degree is 0. Zero has no degree. Find the degree of each monomial a) ¾xdegree: 1 ¾x = ¾x 1. The exponent is 1. b) 7x 2 y 3 degree: 5 The exponents are 2 and 3. Their sum is 5. c) -4degree: 0 The degree of a nonzero constant is 0.

10 Here’s a polynomial 2x 3 – 5x 2 + x + 9 Each one of the little product things is a “term”. 2x 3 – 5x 2 + x + 9 So, this guy has 4 terms. 2x 3 – 5x 2 + x + 9 The coefficients are the numbers in front of the letters. 2x 3 - 5x 2 + x + 9 term 2519 We just pretend this last guy has a letter behind him. Remember x = 1 · x NEXT

11 Since “poly” means many, when there is only one term, it’s a monomial: 5x When there are two terms, it’s a binomial: 2x + 3 When there are three terms, it a trinomial: x 2 – x – 6 So, what about four terms? Quadnomial? Naw, we won’t go there, too hard to pronounce. This guy is just called a polynomial: 7x 3 + 5x 2 – 2x + 4 NEXT

12 So, there’s one word to remember to classify: degree Here’s how you find the degree of a polynomial: Look at each term, whoever has the most letters wins! 3x 2 – 8x 4 + x 5 This is a 7 th degree polynomial: 6mn 2 + m 3 n This guy has 5 letters… The degree is 5. This guy has 7 letters… The degree is 7 NEXT

13 This is a 1 st degree polynomial 3x – 2 What about this dude? 8 How many letters does he have? ZERO! So, he’s a zero degree polynomial This guy has 1 letter… The degree is 1. This guy has no letters… The degree is 0. By the way, the coefficients don’t have anything to do with the degree. Before we go, I want you to know that Algebra isn’t going to be just a bunch of weird words that you don’t understand. I just needed to start with some vocabulary so you’d know what the heck we’re talking about!

14 3x 4 + 5x 2 – 7x + 1 The polynomial above is in standard form. Standard form of a polynomial - means that the degrees of its monomial terms decrease from left to right. term Once you simplify a polynomial by combining like terms, you can name the polynomial based on degree or number of monomials it contains.

15 Classifying Polynomials Write each polynomial in standard form. Then name each polynomial based on its degree and the number of terms. a) 5 – 2x -2x + 5 Place terms in order. linear binomial b) 3x 4 – 4 + 2x 2 + 5x 4 Place terms in order. 3x 4 + 5x 4 + 2x 2 – 4 Combine like terms. 8x 4 + 2x 2 – 4 4 th degree trinomial

16 Write each polynomial in standard form. Then name each polynomial based on its degree and the number of terms. a) 6x – 9x 4 b) 3y – 4 – y 3 c) 8 + 7v – 11v

17 Adding and Subtracting Polynomials The sum or difference

18 Just as you can perform operations on integers, you can perform operations on polynomials. You can add polynomials using two methods. Which one will you choose? Closure of polynomials under addition or subtraction The sum of two polynomials is a polynomial. The difference of two polynomials is a polynomial.

19 Addition of Polynomials Method 1 (vertically) Line up like terms. Then add the coefficients. 4x 2 + 6x + 7-2x 3 + 2x 2 – 5x + 3 2x 2 – 9x x 2 + 4x - 5 6x 2 – 3x + 8-2x 3 + 7x 2 – x - 2 Method 2 (horizontally) Group like terms. Then add the coefficients. (4x 2 + 6x + 7) + (2x 2 – 9x + 1) = (4x 2 + 2x 2 ) + (6x – 9x) + (7 + 1) = 6x 2 – 3x + 8 Example 2: (-2x 3 + 0) + (2x 2 + 5x 2 ) + (-5x + 4x) + (3 – 5) You can rewrite each polynomial, inserting a zero placeholder for the “missing” term. Example 2 Use a zero placeholder

20 Simplify each sum (12m 2 + 4) + (8m 2 + 5) (t 2 – 6) + (3t ) (9w 3 + 8w 2 ) + (7w 3 + 4) (2p 3 + 6p p) + (9p p 2 + 3p ) Remember Use a zero as a placeholder for the “missing” term. Word Problem

21 Find the perimeter of each figure 9c c x - 6 5x + 1 9x 8x - 2 Recall that the perimeter of a figure is the sum of all the sides.

22 Subtracting Polynomials Earlier you learned that subtraction means to add the opposite. So when you subtract a polynomial, change the signs of each of the terms to its opposite. Then add the coefficients. Method 1 (vertically) Line up like terms. Change the signs of the second polynomial, then add. Simplify (2x 3 + 5x 2 – 3x) – (x 3 – 8x ) 2x 3 + 5x 2 – 3x -(x 3 – 8x ) -x 3 + 8x x 3 +13x 2 – 3x - 11 Remember, subtraction is adding the opposite. Method 2

23 Method 2 (horizontally) Simplify (2x 3 + 5x 2 – 3x) – (x 3 – 8x ) Write the opposite of each term. 2x 3 + 5x 2 – 3x – x 3 + 8x 2 – 11 Group like terms. (2x 3 – x 3 ) + (5x 2 + 8x 2 ) + (3x + 0) + ( ) = x x 2 + 3x - 11 = x x 2 + 3x - 11

24 Simplify each subtraction (17n 4 + 2n 3 ) – (10n 4 + n 3 ) (24x x) – (9x x) 6c – 5 2b + 6 7h 2 + 4h - 8 -(4c + 9) -(b + 5) -(3h 2 – 2h + 10)

25 Multiplying and Factoring Using the Distributive Property

26 Observe the rectangle below. Remember that the area A of a rectangle with length l and width w is A = lw. So the area of this rectangle is (4x)(2x), as shown. **************************** The rectangle above shows the example that 4x = x + x + x + x and 2x = x + x 4x 2x A = lw A = (4x)(2x) x + x + x + x x+xx+x NEXT

27 We can further divide the rectangle into squares with side lengths of x. x + x + x + x x+xx+x x2x2 x2x2 x2x2 x2x2 x2x2 x2x2 x2x2 x2x2 x+xx+x Since each side of the squares are x, then x · x = x 2 By applying the area formula for a rectangle, the area of the rectangle must be (4x)(2x). This geometric model suggests the following algebraic method for simplifying the product of (4x)(2x). (4x)(2x) = (4 · x)(2 · x) = (4 · 2)(x · x) = 8x 2 NEXT Commutative Property Associative Property

28 To simplify a product of monomials (4x)(2x) Use the Commutative and Associative Properties of Multiplication to group the numerical coefficients and to group like variable; Calculate the product of the numerical coefficients; and Use the properties of exponents to simplify the variable product. Therefore (4x)(2x) = 8x 2 (4x)(2x) = (4 · 2)(x · x ) = (4 · 2) = 8 (x · x) = x 1 · x 1 = x 1+1 = x 2

29 You can also use the Distributive Property for multiplying powers with the same base when multiplying a polynomial by a monomial. Simplify -4y 2 (5y 4 – 3y 2 + 2) -4y 2 (5y 4 – 3y 2 + 2) = -4y 2 (5y 4 ) – 4y 2 (-3y 2 ) – 4y 2 (2) = Use the Distributive Property -20y y – 8y 2 = Multiply the coefficients and add the -20y y 4 – 8y 2 exponents of powers with the same base. Remember, Multiply powers with the same base: 3 5 · 3 4 = = 3 9

30 Simplify each product. a) 4b(5b 2 + b + 6) b) -7h(3h 2 – 8h – 1) c) 2x(x 2 – 6x + 5) d) 4y 2 (9y 3 + 8y 2 – 11) Remember, Multiplying powers with the same base.

31 Factoring a Monomial from a Polynomial Factoring a polynomial reverses the multiplication process. To factor a monomial from a polynomial, first find the greatest common factor (GCF) of its terms. Find the GCF of the terms of: 4x x 2 – 8x List the prime factors of each term. 4x 3 = 2 · 2 · x · x x 12x 2 = 2 · 2 · 3 · x · x 8x = 2 · 2 · 2 · x The GCF is 2 · 2 · x or 4x.

32 Find the GCF of the terms of each polynomial. a) 5v v 3 b) 3t 2 – 18 c) 4b 3 – 2b 2 – 6b d) 2x x 2 – 6x

33 Factoring Out a Monomial Factor 3x 3 – 12x x Step 1 Find the GCF 3x 3 = 3 · x · x · x 12x 2 = 2 · 2 · 3 · x · x 15x = 3 · 5 · x The GCF is 3 · x or 3x Step 2 Factor out the GCF 3x 3 – 12x x = 3x(x 2 ) + 3x(-4x) + 3x(5) = 3x(x 2 – 4x + 5) To factor a polynomial completely, you must factor until there are no common factors other than 1.

34 Use the GCF to factor each polynomial. a) 8x 2 – 12x b) 5d d c) 6m 3 – 12m 2 – 24m d) 4x 3 – 8x x Try to factor mentally by scanning the coefficients of each term to find the GCF. Next, scan for the least power of the variable.

35 Multiplying Binomials Using the infamous FOIL method

36 Using the Distributive Property As with the other examples we have seen, we can also use the Distributive Property to find the product of two binomials. Simplify: (2x + 3)(x + 4) (2x + 3)(x + 4) = 2x(x + 4) + 3(x + 4) = 2x 2 + 8x + 3x + 12 = 2x x + 12 Now Distribute 2x and 3 Distribute x + 4

37 Simplify each product. a) (6h – 7)(2h + 3) b) (5m + 2)(8m – 1) c) (9a – 8)(7a + 4) d) (2y – 3)(y + 2)

38 Multiplying using FOIL Another way to organize multiplying two binomials is to use FOIL, which stands for, “First, Outer, Inner, Last”. The term FOIL is a memory device for applying the Distributive Property to the product of two binomials. Simplify (3x – 5)(2x + 7) First Outer Inner Last = (3x)(2x) + (3x)(7) – (5)(2x) – (5)(7) (3x – 5)(2x + 7) = 6x x - 10x - 35 = 6x x - 35 The product is 6x x - 35

39 Simplify each product using FOIL a) (3x + 4)(2x + 5) b) (3x – 4)(2x + 5) c) (3x + 4)(2x – 5) d) (3x – 4)(2x – 5) Remember, First, Outer, Inner, Last

40 Applying Multiplication of Polynomials. Find the area of the shaded (beige) region. Simplify. area of outer rectangle = (2x + 5)(3x + 1) area of orange rectangle = x(x + 2) area of shaded region = area of outer rectangle – area of orange portion (2x + 5)(3x + 1) – x(x + 2) = 6x x + 2x + 5 – x 2 – 2x = 6x 2 – x x + 2x – 2x + 5 = 5x x + 5 2x + 5 x + 2 x 3x + 1 Use the FOIL method to simplify (2x + 5)(3x + 1) Use the Distributive Property to simplify –x(x + 2)

41 Find the area of the shaded region. Simplify. Find the area of the green shaded region. Simplify. 5x + 8 6x + 2 5x x + 6

42 FOIL works when you are multiplying two binomials. However, it does not work when multiplying a trinomial and a binomial. ( You can use the vertical or horizontal method to distribute each term.) Simplify (4x 2 + x – 6)(2x – 3) Method 1 (vertical) 4x 2 + x - 6 2x x 2 - 3x + 18 Multiply by -3 8x 3 + 2x x Multiply by 2x 8x x x + 18 Add like terms Remember multiplying whole numbers. 312 x

43 Multiply using the horizontal method. (2x – 3)(4x 2 + x – 6) = 2x(4x 2 ) + 2x(x) + 2x(-6) – 3(4x2) – 3(x) – 3(-6) = 8x 3 + 2x 2 – 12x – 12x 2 – 3x + 18 = 8x 3 -10x x + 18 The product is 8x 3 – 10x 2 – 15x + 18 Drawing arrows between terms can help you identify all six products. Method 2 (horizontal)

44 Simplify using the Distributive Property. a) (x + 2)(x + 5) b) (2y + 5)(y – 3) c) (h + 3)(h + 4) Simplify using FOIL. a) (r + 6)(r – 4) b) (y + 4)(5y – 8) c) (x – 7)(x + 9) WORD PROBLEM

45 Find the area of the green shaded region. x + 3 x - 3 x x + 2


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