Chapter 5 : Chapter 5 Polynomials- Sums, Products, Quotients
Lesson1: Definitions, Addition : Lesson1: Definitions, Addition Degree of the polynomial- the exponent n, where n is the greatest exponent in an expression with one variable or the exponent n where n is the sum of the exponents of the variables in each term.
Like Terms- Terms that contain the same variables raised to the same power.
let’s look at the examples in the book together on page 150 and 151
Lesson 2: Products: (a+b)^2, (a-b)^2, (a+b)^3, (a-b)^3 : Lesson 2: Products: (a+b)^2, (a-b)^2, (a+b)^3, (a-b)^3 Expanded Form- an algebraic expression written to sow its smallest terms. a^2+a^3=a*a*a*a*a
Example on page 155:
Find (x+y)^3
this can be written as (x+y)(x+y)^2
=(x+y)(x^2+2xy+y^2) this is (x+y)^2 written in expanded form, you simply multiply the numbers and variables together using foil.
now you must multiply the expanded expression by each variable in the (x+y)
=x(x^2+2xy+y^2)+y(x^2+2xy+y^2)
now actually do the multiplication
x^3+2x^2y+xy^2+x^2y+2xy^2+y^3
now combine like terms
x^3+3x^2y+3xy^2+y^3
Lesson 3: Factoring a^2-b^2, a^3+b^3, and a^3-b^3 : Lesson 3: Factoring a^2-b^2, a^3+b^3, and a^3-b^3 Examples:
Find the Factors of x^2-y^2
use the model a^2-b^2=(a+b)(a-b)
let a=x and b=y then x^2-y^2=(x+y)(x-y)
Find factors of 9z^2-16
use the a^2-b^2=(a+b)(a-b) model
let a=3z and b=4 (you must find two numbers that are the same and when multiplied equal the original expression. 3z*3z= 9z^2 and 4*4= 16
then 9z^2-16= (3z+4)(3z-4)
Lesson 4: Multiplication of Polynomials : Lesson 4: Multiplication of Polynomials You can use the distributive law to multiply any two polynomials together.
(x+2y)(x^3+5x-7)
the distributive property asks for you to first multiply the larger polynomial by one variable in the smaller and then the other.
x(x^3+5x-7)+2y(x^3+5x-7)
x^4+5x^2-7x+2x^3y+10xy-14y
combine like terms then rearrange the polynomial according to decreasing degree. in this particular problem, there are no like terms.
answer is the same as above but rearranged like so: x^4+2x^3y+5x^2-7x+10xy-147
Lesson 5: Division of Polynomials, Rational Expressions : Lesson 5: Division of Polynomials, Rational Expressions Rational Expression- an algebraic expression divided by another algebraic expression
Find the quotient of 3x^3/x^2
it’s easiest to write out the expressions in expanded form so 3x^3/x^2= 3x*x*x/x*x, you can cancel out numbers in ther denominator as long as they are also in the numerator and only to the extent that for every variable cancelled on the denominator, there must be one variable able to be cancelled in the numerator. in this case there are two single x’s in the numerator and 2 single x’s in the denominator so the can all be cancelled which leaves you with 3x.
Find quotient of (x+y)^3/(x+y)^2= (x+y)(x+y)(x+y)/(x+y)(x+y) the bolds are cancellable so you are left with (x+y) as your answer.
Long Division of Polynomials : Long Division of Polynomials Divide a^3+2a^2b+2ab^2+b^3 by a+b
let’s look at the example on page 165 together and i’m going to write it out for you guys in the white space below for you to see.
view the rules on page 167 on your own
Lesson 7: Complex Fractions : Lesson 7: Complex Fractions Complex fraction- fraction whose numerator, denominator, or both are also fractions.
Reciprocal Fractions- for any fraction a/b the fraction b/a such as (a/b)*(b/a)=1
Find 1/2 divided by 2/3 or (1/2)/(2/3) multiply the top and bottom by the reciprocal of the denominator. so (1/2)*(3/2)/(2/3)*(3/2)
your denominator according to the definition of reciprocal fractions should equal 1. your numerator1/2*3/2= (1*3)/(2*2)=3/4 so you have (3/4)/1= 3/4 is your answer
Lesson 8: Geometry Connection: Perimeter Formula : Lesson 8: Geometry Connection: Perimeter Formula Perimeter- distance around a geometric figure.
view the examples on your own.