Presentation on theme: "Polynomials and Factoring"— Presentation transcript:
1Polynomials and Factoring The basic building blocks of algebraic expressions
2The 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) (5) + 10 = feet
3In regular math books, this is called “substituting” or “evaluating”… We are given the algebraic expression below and asked to evaluate it.x2 – 4x + 1We need to find what this equals when we put a number in for x.. Likex = 3Everywhere you see an x… stick in a 3!= (3)2 – 4(3) + 1= 9 –= -2
4You try a couple Use the same expression but let x = 2 and x = -1 What about x = -5?Be careful with the negative! Use ( )!x2 – 4x + 1= (-5)2 – 4(-5) + 1= 46You try a couple Use the same expression but let x = 2 and x = -1
5That critter in the last slide is a polynomial. x2 – 4x + 1 Here are some othersx2 + 7x – 34a3 + 7a2 + anm2 – m3x – 25
6For 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.
7Some 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
8The ending of these words “nomial” is Greek for “part”. 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 difference (-) of one or more terms. (poly implies many) x2 + 2x, 3x3 + x2 + 5x + 6, 4x + 6y + 8The ending of these words “nomial” is Greek for “part”.Polynomials are in simplest form when they contain no like terms. x2 + 2x x2 – 4x when simplified becomes 4x2 – 2x + 1Polynomials 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.
9The 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 monomiala) 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.
10We just pretend this last guy has a letter behind him. Here’s a polynomial 2x3 – 5x2 + x + 9 Each one of the little product things is a “term”. 2x3 – 5x x So, this guy has 4 terms. 2x3 – 5x2 + x The coefficients are the numbers in front of the letters. 2x x2 + x + 9termtermtermtermNEXT2519We just pretend this last guy has a letter behind him.Rememberx = 1 · x
11Since “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 + 4NEXT
12So, 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 is a 7th degree polynomial: 6mn2 + m3n4 + 8This guy has 5 letters…The degree is 5.This guy has 7 letters…The degree is 7NEXT
13This guy has no letters… This is a 1st degree polynomial 3x – 2 What about this dude? 8 How many letters does he have? ZERO! So, he’s a zero degree polynomialThis 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!
143x4 + 5x2 – 7x + 1 The polynomial above is in standard form 3x4 + 5x2 – 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.termtermtermtermOnce you simplify a polynomial by combining like terms, you can name the polynomial based on degree or number of monomials it contains.
15Classifying 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 binomialb) 3x4 – 4 + 2x2 + 5x4 Place terms in order.3x4 + 5x4 + 2x2 – 4 Combine like terms.8x4 + 2x2 – 44th degree trinomial
16Write each polynomial in standard form 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) v – 11v
17Adding and Subtracting Polynomials The sum or difference
18Closure of polynomials under addition or subtraction 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 subtractionThe sum of two polynomials is a polynomial.The difference of two polynomials is a polynomial.
19Addition 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 x3 + 2x2 – 5x + 32x2 – 9x x2 + 4x - 56x2 – 3x x3 + 7x2 – x - 2Method 2 (horizontally)Group like terms. Then add the coefficients.(4x2 + 6x + 7) + (2x2 – 9x + 1) = (4x2 + 2x2) + (6x – 9x) + (7 + 1)= 6x2 – 3x + 8Example 2:(-2x3 + 0) + (2x2 + 5x2) + (-5x + 4x) + (3 – 5)Example 2Use a zero placeholder
20Use a zero as a placeholder for the “missing” term. Simplify each sum(12m2 + 4) + (8m2 + 5)(t2 – 6) + (3t2 + 11)(9w3 + 8w2) + (7w3 + 4)(2p3 + 6p2 + 10p) + (9p3 + 11p2 + 3p )RememberUse a zero as a placeholder for the “missing” term.Word Problem
21Find the perimeter of each figure 17x - 68x - 25c + 25x + 19xRecall that the perimeter of a figure is the sum of all the sides.
22Subtracting 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 – 8x ) -x3 + 8xx3 +13x2 – 3x - 11Remember, subtraction is adding the opposite.Method 2
23Method 2 (horizontally) Simplify (2x3 + 5x2 – 3x) – (x3 – 8x2 + 11) Write the opposite of each term. 2x3 + 5x2 – 3x – x3 + 8x2 – 11 Group like terms. (2x3 – x3) + (5x2 + 8x2) + (3x + 0) + ( ) = x x x = x3 + 13x2 + 3x - 11
25Multiplying 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 that4x = x + x + x + x and 2x = x + x4x2xA = lwA = (4x)(2x)x + x + x + xx+xNEXT
27Since each side of the squares are x, then x · x = x2 We can further divide the rectangle into squares with side lengths of x.x + x + x + xx+xSince each side of the squares are x, then x · x = x2x + x + x + xx+xx2x2x2x2x2x2x2x2By 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) = 8x2NEXTCommutative PropertyAssociative Property
28To 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; andUse 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
29Multiply powers with the same base: 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) = y2(5y4) – 4y2(-3y2) – 4y2(2) = Use the Distributive Property -20y y2 + 2 – 8y2 = Multiply the coefficients and add the -20y6 + 12y4 – 8y exponents of powers with the same base.Remember,Multiply powers with the same base:35 · 34 = = 39
30Multiplying powers with the same base. 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.
31Factoring 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:4x3 + 12x2 – 8xList the prime factors of each term.4x3 = 2 · 2 · x · x x12x2 = 2 · 2 · 3 · x · x8x = 2 · 2 · 2 · xThe GCF is 2 · 2 · x or 4x.
32Find the GCF of the terms of each polynomial Find the GCF of the terms of each polynomial. a) 5v5 + 10v3 b) 3t2 – 18 c) 4b3 – 2b2 – 6b d) 2x4 + 10x2 – 6x
33Factoring Out a Monomial To factor a polynomial completely, you must factor until there are no common factors other than 1.Factor 3x3 – 12x2 + 15xStep 1Find the GCF3x3 = 3 · x · x · x12x2 = 2 · 2 · 3 · x · x15x = 3 · 5 · xThe GCF is 3 · x or 3xStep 2Factor out the GCF3x3 – 12x2 + 15x= 3x(x2) + 3x(-4x) + 3x(5)= 3x(x2 – 4x + 5)
34Use the GCF to factor each polynomial Use the GCF to factor each polynomial. a) 8x2 – 12x b) 5d3 + 10d c) 6m3 – 12m2 – 24m d) 4x3 – 8x2 + 12xTry to factor mentally by scanning the coefficients of each term to find the GCF. Next, scan for the least power of the variable.
35Multiplying Binomials Using the infamous FOIL method
36Using the Distributive Property Distribute x + 4As 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 + 12Now Distribute 2x and 3
38Multiplying 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 x= 6x xThe product is 6x2 + 11x - 35
40Applying Multiplication of Polynomials. 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) =6x2 + 15x + 2x + 5 – x2 – 2x =6x2 – x2 + 15x + 2x – 2x =5x2 + 17x + 5Find the area of the shaded (beige) region. Simplify.2x + 5x + 23x + 1xUse the Distributive Property to simplify –x(x + 2)Use the FOIL method to simplify (2x + 5)(3x + 1)
41Find the area of the shaded region. Simplify. Find the area of the green shaded region. Simplify.5x + 86x + 25xx + 6
42Remember multiplying whole numbers. 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.312x 239366247176Simplify (4x2 + x – 6)(2x – 3)Method 1 (vertical)4x2 + x - 62x - 3-12x2 - 3x Multiply by -38x x x Multiply by 2x8x x x + 18 Add like terms
43Multiply 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)= 8x x2 – 12x – 12x2 – 3x= 8x3 -10x xThe product is 8x3 – 10x2 – 15x + 18
44Simplify using the Distributive Property 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
45Find the area of the green shaded region. x + 3xx + 2x - 3