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Polynomials and Factoring

Polynomials and Factoring. The basic building blocks of algebraic expressions.

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Polynomials and Factoring

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  1. Polynomials and Factoring The basic building blocks of algebraic expressions

  2. The height in feet ofa fireworks launched straight up into the airfrom (s) feet off the ground at velocity (v)after (t) seconds is given by the equation:-16t2 + vt + sFind 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 + 10610 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. You try a coupleUse the same expression but letx = 2 andx = -1 What about x = -5? Be careful with the negative! Use ( )! x2 – 4x + 1 = (-5)2 – 4(-5) + 1 = 46

  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 + 2polynomial – is a monomial or the sum (+) or difference (-) of one or more terms. (poly implies many).x2 + 2x, 3x3 + x2 + 5x + 6, 4x + 6y + 8 The ending of these words “nomial” is Greek for “part”. • 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 x0.

  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 polynomial2x3 – 5x2 + x + 9Each one of the little product things is a “term”.2x3 – 5x2 + x + 9So, this guy has 4 terms.2x3 – 5x2 + x + 9The coefficients are the numbers in front of the letters.2x3 - 5x2 + x + 9 term term term term NEXT 2 5 1 9 We just pretend this last guy has a letter behind him. Remember x = 1 · x

  11. Since “poly” means many, when there is only one term, it’s a monomial:5xWhen there are two terms, it’s a binomial:2x + 3When there are three terms, it a trinomial:x2 – x – 6So, 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:degreeHere’s how you find the degree of a polynomial:Look at each term,whoever has the most letters wins!3x2 – 8x4 + x5This is a 7th degree polynomial:6mn2 + m3n4 + 8 This guy has 5letters… The degree is 5. This guy has 7letters… The degree is 7 NEXT

  13. This is a 1st degree polynomial3x – 2What about this dude?8How many letters does he have? ZERO!So, he’s a zero degree polynomial This guy has 1 letter… The degree is 1. By the way, the coefficients don’t have anything to do with the degree. This guy has no letters… The degree is 0. 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 + 1The 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 term term 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 + 5Place terms in order. linear binomial b) 3x4 – 4 + 2x2 + 5x4Place terms in order. 3x4 + 5x4 + 2x2 – 4 Combine like terms. 8x4 + 2x2 – 4 4th degree trinomial

  16. Write each polynomial in standardform. Then name each polynomial based on its degree and the number of terms.a) 6x2 + 7 – 9x4b) 3y – 4 – y3c) 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 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 + 10 + 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) • (9w3 + 8w2) + (7w3 + 4) • (2p3 + 6p2 + 10p) + (9p3 + 11p2 + 3p ) Remember Use a zero as a placeholder for the “missing” term. Word Problem

  21. Find the perimeter of each figure 9c - 10 17x - 6 8x - 2 5c + 2 5x + 1 9x 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

  23. Method 2 (horizontally)Simplify (2x3 + 5x2 – 3x) – (x3 – 8x2 + 11)Write the opposite of each term.2x3 + 5x2 – 3x – x3+ 8x2– 11Group like terms.(2x3 – x3) + (5x2 + 8x2) + (3x + 0) + (-11 + 0) = x3 + 13x2 + 3x - 11 = x3 + 13x2 + 3x - 11

  24. Simplify each subtraction • (17n4 + 2n3) – (10n4 + n3) • (24x5 + 12x) – (9x5 + 11x) • 6c – 5 2b + 6 7h2 + 4h - 8 -(4c + 9)-(b + 5)-(3h2 – 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+x NEXT

  27. 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+x x2 x2 x2 x2 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) = 8x2 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) = 8x2 (4x)(2x) = (4 · 2)(x · x ) = (4 · 2) =8 (x · x) = x1 · x1 = x1+1 = x2

  29. 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)-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 – 8y2exponents of powers with the same base. Remember, Multiply powers with the same base: 35 · 34 = 35 + 4 = 39

  30. Simplify each product.a) 4b(5b2 + b + 6)b) -7h(3h2 – 8h – 1)c) 2x(x2 – 6x + 5)d) 4y2(9y3 + 8y2 – 11) Remember, Multiplying powers with the same base.

  31. Factoring a polynomial reverses the multiplication process. To factor a monomial from a polynomial, first find the greatest commonfactor (GCF) of its terms. Find the GCF of the terms of: 4x3 + 12x2 – 8x List the prime factors of each term. 4x3 = 2 · 2 · x · x x 12x2 = 2 · 2 · 3 · x · x 8x = 2 · 2 · 2 · x The GCF is 2 · 2 · x or 4x. Factoring a Monomial from a Polynomial

  32. Find the GCF of the terms of each polynomial.a) 5v5 + 10v3b) 3t2 – 18c) 4b3 – 2b2 – 6bd) 2x4 + 10x2 – 6x

  33. Factor 3x3 – 12x2 + 15x Step 1 Find the GCF 3x3 = 3 · x · x · x 12x2 = 2 · 2 · 3 · x · x 15x = 3 · 5 · x The GCF is 3 · x or 3x Step 2 Factor out the GCF 3x3 – 12x2 + 15x = 3x(x2) + 3x(-4x) + 3x(5) = 3x(x2 – 4x + 5) Factoring Out a Monomial 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) 8x2 – 12xb) 5d3 + 10dc) 6m3 – 12m2 – 24md) 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.

  35. Multiplying Binomials Using the infamous FOIL method

  36. 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) = 2x2 + 8x + 3x + 12 = 2x2 + 11x + 12 Using the Distributive Property Distribute x + 4 Now Distribute 2x and 3

  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. 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) FirstOuterInnerLast = (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 Multiplying using FOIL

  39. 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)

  40. 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 oforange portion (2x + 5)(3x + 1) – x(x + 2) = 6x2 + 15x + 2x + 5 – x2 – 2x = 6x2 – x2 + 15x + 2x – 2x + 5 = 5x2 + 17x + 5 Applying Multiplication of Polynomials. 2x + 5 x + 2 3x + 1 x Use the Distributive Property to simplify –x(x + 2) Use the FOIL method to simplify (2x + 5)(3x + 1)

  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.) Remember multiplying whole numbers. 312 x 23 936 624 7176 Simplify (4x2 + x – 6)(2x – 3) Method 1 (vertical) 4x2 + x - 6 2x - 3 -12x2 - 3x + 18 Multiply by -3 8x3 + 2x2 - 12x Multiply by 2x 8x3 - 10x2 - 15x + 18 Add like terms

  43. Multiply using the horizontal method. Drawing arrows between terms can help you identify all six products. Method 2 (horizontal) (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

  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 x + 2 x - 3

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