Polynomials and Factoring
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)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.
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
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!
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 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.
Here’s a polynomial 2x3 – 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
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
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 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
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.
Classifying Polynomials
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
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?
Addition of Polynomials
Simplify each sum
Find the perimeter of each figure
Subtracting Polynomials
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
Slide 24
Simplify each subtraction
Multiplying and Factoring
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.
We can further divide the rectangle into squares with side lengths of x.
To simplify a product of monomials (4x)(2x)
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.
Simplify each product. a) 4b(5b2 + b + 6) b) -7h(3h2 – 8h – 1) c) 2x(x2 – 6x + 5) d) 4y2(9y3 + 8y2 – 11)
Factoring a Monomial from a 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
Use the GCF to factor each polynomial. a) 8x2 – 12x b) 5d3 + 10d c) 6m3 – 12m2 – 24m d) 4x3 – 8x2 + 12x
Multiplying Binomials
Using the Distributive Property
Simplify each product.
Multiplying using FOIL
Simplify each product using FOIL
Applying Multiplication of Polynomials.
Find the area of the shaded region. Simplify.
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.)
Multiply using the horizontal method.
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)
Find the area of the green shaded region.