This pdf includes the following topics:- Polynomials and Factoring monomial binomial trinomial polynomial degree of a monomial Adding and Subtracting Polynomials and many more.
1. Polynomials Polynomials and and Factoring 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: -16t2 + vt + s Find the height of a firework launched from a 10 ft platform at 200 ft/s after 5 seconds. -16t2 + vt + s -16(5)2 + 200(5) + 10 =400 + 1600 + 10 610 feet
3. In regular math books, this is called “substituting” or “evaluating”… We are given the algebraic expression below and asked to evaluate it. x2 – 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! x2 – 4x + 1 = (3)2 – 4(3) + 1 = 9 – 12 + 1 = -2
4. What about x = -5? Be careful with the negative! Use ( )! x2 – 4x + 1 = (-5)2 – 4(-5) + 1 = 46 You try a couple Use the same expression but let x = 2 and x = -1
5. That critter in the last slide is a polynomial. x2 – 4x + 1 Here are some others x2 + 7x – 3 4a3 + 7a2 + a nm2 – 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, x2, 4y2, -2xy, or 520x2y2 (notice: no negative exponents, no fractional exponents) binomial – is the sum of two monomials. It has two unlike terms (bi implies two). 3x + 1, x2 – 4x, 2x + y, or y – y2
8. trinomial – is the sum of three monomials. It has three unlike terms. (tri implies three). x2 + 2x + 1, 3x2 – 4x + 10, 2x + 3y + 2 polynomial – is a monomial or the sum (+) or The ending of these words “nomial” is Greek difference (-) of one or more terms. for “part”. (poly implies many). x2 + 2x, 3x3 + x2 + 5x + 6, 4x + 6y + 8 • Polynomials are in simplest form when they contain no like terms. x2 + 2x + 1 + 3x2 – 4x when simplified becomes 4x2 – 2x + 1 • Polynomials are generally written in descending order. Descending: 4x2 – 2x + 1 (exponents of variables decrease from left to right) Constants like 12 are monomials since they can be written as 12x0 = 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) ¾x degree: 1 ¾x = ¾x1. The exponent is 1. b) 7x2y3 degree: 5 The exponents are 2 and 3. Their sum is 5. c) -4 degree: 0 The degree of a nonzero constant is 0.
10. Here’s a polynomial 2x3 – 5x2 + x + 9 Each one of the little product things is a “term”. 2x3 – 5x2 + x + 9 term term term term So, this guy has 4 terms. 2x3 – 5x2 + x + 9 The coefficients are the numbers in front of the letters. 2x3 - 5x2 + x + 9 NEXT 2 5 1 9 We just pretend this last guy has a Remember letter behind him. x=1·x
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: x2 – 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: 7x3 + 5x2 – 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! 3x2 – 8x4 + x5 This guy has 5 letters… The degree is 5. This is a 7th degree polynomial: 6mn2 + m3n4 + 8 This guy has 7 letters… The degree is 7 NEXT
13. This is a 1st degree polynomial 3x – 2 This guy has 1 letter… The degree is 1. By the way, the coefficients don’t have anything to What about this dude? This guy has no do with the degree. 8 letters… The degree is 0. How many letters does he have? ZERO! So, he’s a zero degree polynomial 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. 3x4 + 5x2 – 7x + 1 term term term term 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. Name using Polynomial Degree Name using Number of number of Degree Terms terms 7x + 4 1 Linear 2 Binomial 2 3x + 2x + 1 2 Quadratic 3 Trinomial 4x3 3 Cubic 1 Monomial 9x4 + 11x 4 Fourth degree 2 Binomial 5 0 Constant 1 monomial 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) 3x4 – 4 + 2x2 + 5x4 Place terms in order. 3x4 + 5x4 + 2x2 – 4 Combine like terms. 8x4 + 2x2 – 4 4th degree trinomial
16. Write each polynomial in standard form. Then name each polynomial based on its degree and the number of terms. a) 6x2 + 7 – 9x4 b) 3y – 4 – y3 c) 8 + 7v – 11v
17. Adding and and Subtracting Subtracting Polynomials 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 You can rewrite each polynomial, inserting a zero placeholder for the “missing” term. Method 1 (vertically) Line up like terms. Then add the coefficients. 4x2 + 6x + 7 -2x3 + 2x2 – 5x + 3 2x2 – 9x + 1 0 + 5x2 + 4x - 5 6x2 – 3x + 8 -2x3 + 7x2 – x - 2 Method 2 (horizontally) Group like terms. Then add the coefficients. (4x2 + 6x + 7) + (2x2 – 9x + 1) = (4x2 + 2x2) + (6x – 9x) + (7 + 1) = 6x2 – 3x + 8 Example 2: (-2x3 + 0) + (2x2 + 5x2) + (-5x + 4x) + (3 – 5) Example 2 Use a zero placeholder
20. Simplify each sum • (12m2 + 4) + (8m2 + 5) • (t2 – 6) + (3t2 + 11) Remember Use a zero as a placeholder for the “missing” term. • (9w3 + 8w2) + (7w3 + 4) • (2p3 + 6p2 + 10p) + (9p3 + 11p2 + 3p ) Word Problem
21. Find the perimeter of each figure 9c - 10 - 6 17x 8x - 2 5c + 2 5x 9x + 1 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 (2x3 + 5x2 – 3x) – (x3 – 8x2 + 11) 2x3 + 5x2 – 3x 2x3 + 5x2 – 3x -(x3 – 8x2 + 0 + 11) -x3 + 8x2 + 0 - 11 x3 +13x2 – 3x - 11 Remember, subtraction is adding the opposite. Method 2
26. Multiplying and and Factoring Factoring Using the Distributive Property
27. 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. 4x 2x A = lw A = (4x)(2x) **************************** x+x+x+x x + x The rectangle above shows the example that 4x = x + x + x + x and 2x = x + x NEXT
28. We can further divide the rectangle into squares with side lengths of x. x +x+x+ x x + x Since each side of the squares are x, then x · x = x2 x+x+x+x x x2 x2 x2 x2 + x x2 x2 x2 x2 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
29. 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; (4x)(2x) = (4 · 2)(x · x ) = • Calculate the product of the numerical coefficients; and (4 · 2) = 8 • Use the properties of exponents to simplify the variable product. (x · x) = x1 · x1 = x1+1 = x2 Therefore (4x)(2x) = 8x2
30. You can also use the Distributive Property for multiplying powers with the same base when multiplying a polynomial by a monomial. Simplify -4y2(5y4 – 3y2 + 2) Remember, Multiply powers with the same base: 35 · 34 = 35 + 4 = 39 -4y2(5y4 – 3y2 + 2) = -4y2(5y4) – 4y2(-3y2) – 4y2(2) = Use the Distributive Property -20y2 + 4 + 12y2 + 2 – 8y2 = Multiply the coefficients and add the -20y6 + 12y4 – 8y2 exponents of powers with the same base.
31. Simplify each product. a) 4b(5b2 + b + 6) b) -7h(3h2 – 8h – 1) c) 2x(x2 – 6x + 5) Remember, Multiplying powers with the same base. d) 4y2(9y3 + 8y2 – 11)
32. Factoring a Monomial from a Polynomial Factoring a polynomial reverses the multiplication process. Find the GCF of the terms of: To factor a monomial 4x3 + 12x2 – 8x from a polynomial, first List the prime factors of each term. find the greatest 4x3 = 2 · 2 · x · x x common factor (GCF) of 12x2 = 2 · 2 · 3 · x · x its terms. 8x = 2 · 2 · 2 · x The GCF is 2 · 2 · x or 4x.
33. Find the GCF of the terms of each polynomial. a) 5v5 + 10v3 b) 3t2 – 18 c) 4b3 – 2b2 – 6b d) 2x4 + 10x2 – 6x
34. Factoring Out a To factor a polynomial Monomial completely, you must factor until there are no common factors other than 1. Factor 3x3 – 12x2 + 15x Step 1 Find the GCF 3x3 = 3 · x · x · x Step 2 12x2 = 2 · 2 · 3 · x · x Factor out the GCF 15x = 3 · 5 · x 3x3 – 12x2 + 15x The GCF is 3 · x or 3x = 3x(x2) + 3x(-4x) + 3x(5) = 3x(x2 – 4x + 5)
35. Use the GCF to factor each polynomial. a) 8x2 – 12x b) 5d3 + 10d c) 6m3 – 12m2 – 24m d) 4x3 – 8x2 + 12x Try to factor mentally by scanning the coefficients of each term to find the GCF. Next, scan for the least power of the variable.
36. Multiplying Binomials Binomials Using the infamous FOIL method
37. Using the Distributive Property Distribute x + 4 As with the other Simplify: (2x + 3)(x + 4) examples we have seen, we can also (2x + 3)(x + 4) = use the Distributive Property to find the 2x(x + 4) + 3(x + 4) = product of two 2x2 + 8x + 3x + 12 = binomials. 2x2 + 11x + 12 Now Distribute 2x and 3
38. Simplify each product. a) (6h – 7)(2h + 3) b) (5m + 2)(8m – 1) c) (9a – 8)(7a + 4) d) (2y – 3)(y + 2)
39. 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) = 6x2 + 21x - 10x - 35 = 6x2 + 11x - 35 The product is 6x2 + 11x - 35
40. Simplify each product using FOIL Remember, First, Outer, Inner, Last a) (3x + 4)(2x + 5) b) (3x – 4)(2x + 5) c) (3x + 4)(2x – 5) d) (3x – 4)(2x – 5)
41. Applying Multiplication of area of outer rectangle = Polynomials. (2x + 5)(3x + 1) area of orange rectangle = Find the area of the x(x + 2) shaded (beige) region. area of shaded region Simplify. = area of outer rectangle – area of 2x + 5 orange portion (2x + 5)(3x + 1) – x(x + 2) = x+2 6x2 + 15x + 2x + 5 – x2 – 2x = 3x + 1 x 6x2 – x2 + 15x + 2x – 2x + 5 = 5x2 + 17x + 5 Use the Distributive Property Use the FOIL method to to simplify –x(x + 2) simplify (2x + 5)(3x + 1)
42. Find the area of the shaded region. Simplify. Find the area of the green shaded region. Simplify. 5x + 8 6x + 2 5x x+6
43. 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.) Remember multiplying Simplify (4x + x – 6)(2x – 3) 2 whole numbers. 312 Method 1 (vertical) x 23 936 4x2 + x - 6 624 7176 2x - 3 -12x2 - 3x + 18 Multiply by -3 8x3 + 2x2 - 12x Multiply by 2x 8x3 - 10x2 - 15x + 18 Add like terms
44. Multiply using the horizontal method. Drawing arrows Method 2 (horizontal) between terms can help you identify all six products. (2x – 3)(4x2 + x – 6) = 2x(4x2) + 2x(x) + 2x(-6) – 3(4x2) – 3(x) – 3(-6) = 8x3 + 2x2 – 12x – 12x2 – 3x + 18 = 8x3 -10x2 - 15x + 18 The product is 8x3 – 10x2 – 15x + 18
45. 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
46. Find the area of the green shaded region. x+3 x+2 x-3 x
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