Polynomials and Factoring

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Polynomials and Factoring
The basic building blocks of algebraic expressions

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) (5) + 10 = feet

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! = (3)2 – 4(3) + 1 = 9 – = -2

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 ( )! x2 – 4x + 1 = (-5)2 – 4(-5) + 1 = 46 You try a couple Use the same expression but let x = 2 and x = -1

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

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.

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

The 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 + 8 The 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 + 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.

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.

We 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 + 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

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

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 is a 7th degree polynomial: 6mn2 + m3n4 + 8 This guy has 5 letters… The degree is 5. This guy has 7 letters… The degree is 7 NEXT

This 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 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!

3x4 + 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. 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.

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

Write 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

The sum or difference

Closure 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 subtraction The sum of two polynomials is a polynomial. The difference of two polynomials is a polynomial.

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 + 3 2x2 – 9x x2 + 4x - 5 6x2 – 3x x3 + 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

Use 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 ) Remember Use a zero as a placeholder for the “missing” term. Word Problem

Find the perimeter of each figure
17x - 6 8x - 2 5c + 2 5x + 1 9x Recall that the perimeter of a figure is the sum of all the sides.

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 – 8x ) -x3 + 8x x3 +13x2 – 3x - 11 Remember, subtraction is adding the opposite. Method 2

Method 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

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

Multiplying and Factoring
Using the Distributive Property

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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

Since 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 + 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

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

Multiply 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

Multiplying 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.

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: 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.

Find 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

Factoring Out a Monomial
To factor a polynomial 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 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)

Use 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 + 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.

Multiplying Binomials
Using the infamous FOIL method

Using the Distributive Property
Distribute x + 4 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 Now Distribute 2x and 3

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

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 x = 6x x The product is 6x2 + 11x - 35

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)

Applying 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 + 5 Find the area of the shaded (beige) region. Simplify. 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)

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

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

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) = 8x x2 – 12x – 12x2 – 3x = 8x3 -10x x The product is 8x3 – 10x2 – 15x + 18

Simplify 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

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