Polynomials and Factoring

The basic building blocks of algebraic expressions

The height in feet ofa 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

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

You try a coupleUse 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

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

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

• 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)

The ending of these words “nomial” is Greek

for “part”.

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.

Here’s a polynomial2x3 – 5x2 + x + 9

Each one of the little product things is a “term”.2x3 – 5x2 + x + 9

So, this guy has 4 terms.2x3 – 5x2 + x + 9

The coefficients are the numbers in front of the letters.2x3 - 5x2 + x + 9

term term term term

2 5 1 9We just pretend this last guy has a letter behind him.

Remember x = 1 · x

NEXT

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 is a 1st degree polynomial3x – 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!

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

termtermterm

Polynomial

Degree

Name using Degree

Number of

Terms

Name using number of

terms 7x + 4 1 Linear 2 Binomial

3x2 + 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.

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. 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

Adding and Subtracting Polynomials

The sum or difference

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.

Addition of Polynomials

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)

You can rewrite each polynomial, inserting a zero placeholder for

the “missing” term.

Example 2 Use a zero placeholder

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

9c - 10

5c + 2

17x - 6

5x + 1 9x

8x - 2

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 – 8x2 + 0 + 11) -x3 + 8x2 + 0 - 11 x3 +13x2 – 3x - 11Remember,

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) + (-11 + 0) = x3 + 13x2 + 3x - 11 = x3 + 13x2 + 3x - 11

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)

Multiplying and Factoring

Using the Distributive Property

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

We can further divide the rectangle into squares with side lengths of x.

x + x + x + x

x+x

x2 x2 x2 x2

x2 x2 x2 x2

x + x + x + x

x+x

Since each side of the squares are x, then x · x = 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

NEXTCommutative 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

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 – 8y2 exponents of powers with the same base.

Remember, Multiply powers with the same base:

35 · 34 = 35 + 4 = 39

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.a) 5v5 + 10v3

b) 3t2 – 18 c) 4b3 – 2b2 – 6bd) 2x4 + 10x2 – 6x

Factoring Out a Monomial

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)

To factor a polynomial completely, you must factor until there are no common

factors other than 1.

Use the GCF to factor each polynomial.a) 8x2 – 12xb) 5d3 + 10d c) 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.

Multiplying BinomialsUsing the infamous FOIL method

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) = 2x2 + 8x + 3x + 12 = 2x2 + 11x + 12

Now Distribute 2x and 3

Distribute x + 4

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) = 6x2 + 21x - 10x - 35 = 6x2 + 11x - 35

The product is 6x2 + 11x - 35

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

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) = 6x2 + 15x + 2x + 5 – x2 – 2x = 6x2 – x2 + 15x + 2x – 2x + 5 = 5x2 + 17x + 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)

Find the area of the shaded region.

Simplify.

Find the area of the green shaded region. Simplify.

5x + 8

6x +

25x

x + 6

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 (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

Remember multiplying whole numbers.

312 x 23 936 624

7176

Multiply using the horizontal method.

(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

Drawing arrows between terms can

help you identify all six products.

Method 2 (horizontal)

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 - 3

x

x + 2