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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
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