POLYNOMIALS€¦ · MULTIPLYING A BINOMIAL BY A BINOMIAL HORIZONTAL (FOIL) When 2 binomials are...
Transcript of POLYNOMIALS€¦ · MULTIPLYING A BINOMIAL BY A BINOMIAL HORIZONTAL (FOIL) When 2 binomials are...
UNIT 6POLYNOMIALS
Polynomial (Definition)
A monomial or a sum of monomials.
A monomial is measured by its degree
To find its degree, we add up the exponents of all the variables of the monomial.
Ex.
2 x3 y4 z5
2 has a degree of 0 (do not mistake degree for zero exponent)
x3 has an exponent of 3
y4 has an exponent of 4
z5 has an exponent of 5
Add up the exponents of your variables only
3 + 4 + 5 = 12
This monomial would be a 12th degree polynomial
Let's Practice Finding the Degree of a Monomial
3 degree of ___
3x degree of ___
3x2 degree of ___
3x3 degree of ___
3x4 degree of ___
DEGREE OF A POLYNOMIAL
The degree of a polynomial is determined by the monomial with the highest degree.
3 would be a 0 degree polynomial because all numbers have a degree of zero. (NOT TO BE CONFUSED WITH ZERO EXPONENT PROPERTY).
We call 0 degree polynomials CONSTANT.
3x would be a 1st degree polynomial because the 3 has a degree of 0 and x has an exponent of 1.
We call 1st degree polynomials LINEAR.
3x2 would be a 2nd degree polynomial because the 3 has a degree of 0 and x2 has an exponent of 2.
We call 2nd degree polynomials QUADRATIC.
3x3 would be a 3rd degree polynomial because the 3 has a degree of 0 and x3 has an exponent of 3.
We call 3rd degree polynomials CUBIC.
3x4 would be a 4th degree polynomial because the 3 has a degree of 0 and x4 has an exponent of 4.
We call 4th degree polynomials QUARTIC.
3x5 would be a 5th degree polynomial because the 3 has a degree of 0 and x5 has an exponent of 5.
We call 5th degree polynomials QUINTIC.
3x6 would be a 6th degree polynomial because the 3 has a degree of 0 and x6 has an exponent of 6.
We call 6th degree and above their degree name, so it would be a polynomial to the 6TH DEGREE.
NAMING A POLYNOMIAL
The number of terms added together determines the name of a polynomial.
6 has only 1 term and would be called a MONOMIAL.
x + 3 has 2 terms and would be called a BINOMIAL.
x2 + x + 3 had 3 terms and would be called a TRINOMIAL.
x3 + x2 + x + 3 has 4 terms and would be called a POLYNOMIAL.
Any polynomial with 4 or more terms is also called a
POLYNOMIAL.
Use this chart as a guide
Let's Practice (Fill out this table)
The terms of a polynomial are usually arranged so that the terms are in order from greatest degree to least degree. This is called the standard form of a polynomial.
STANDARD FORM
Ex:
x2 + x + 1 is in standard form because the terms are arranged from greatest to least degree from left to right.
2nd degree, 1st degree, zero degree
Practice problems 7-12 on page 5 of Ch 8 Sec 1 Study Guide by writing each polynomial in standard form.
Leading Coeffcient
The number in front of the monomial with the largest degree.
Ex.
x2 + x + 1
1 is the leading coefficient since x2 is the highest degree of any of the 3 monomials.
Practice problems 7-12 on page 5 of Ch 8 Sec 1 Study Guide by determining the leading coefficient of each polynomial.
MULTIPLYING A POLYNOMIAL AND A MONOMIAL
METHOD 1 HORIZONTAL
x( 5x + x2 )
x(5x) + x(x2) distribute the x to each term inside the parenthesis
5x2 + x3 simplify
x3 + 5x2 put in standard form (greatest to least degree)
1 is the leading coefficient
Cubic Binomial is its proper name
METHOD 2 VERTICAL
x( 5x + x2 )
5x + x2
(x) x multiply x to each term
5x2 + x3
x3 + 5x2 put in standard form (greatest to least degree)
1 is the leading coefficient
Cubic Binomial is its proper name
Practice multiplying by one of the methods on
Worksheet 8-2 Study Guide page 12 (1-9)
MULTIPLYING A BINOMIAL BY A BINOMIAL HORIZONTAL (FOIL)
When 2 binomials are multiplied together, the result will always result in 4 terms. Those 4 terms will be simplified into standard form and result in a TRINOMIAL.
We will concentrate on Quadratic Trinomials.
Practice using the FOIL method on
8-3 Study Guide page 18 (1-18)
SPECIAL PRODUCTS
Ex. 1 Square of a Sum
( x + 4 )2 a is x b is 4 2(a)(b) is 2(x)(4)
a2 = (x)2
2ab = 2(4x)
b2 = (4)2
Put them all together to get
x2 + 8x + 16
Ex. 2 Square of a difference
( x - 6 )2 a is x b is -6 2(a)(b) is 2(x)(-6)
a2 = (x)2
2ab = 2(-6x)
b2 = (-6)2
Put them all together to get
x2 -12x + 36
Practice solving for special products on
8-3 Study Guide page 25 (1-21)
Ex. 3
( x + 4 ) ( x - 4 ) a is x b is 4 and -4
2ab is 2(x)(4) and 2(x)(-4)
a2 = (x)2
2ab = 2(4x) + 2(-4x)
b2 = -(4)2
Put them all together to get
x2 + 8x -8x -16 Simplify
x2 -16
Practice finding special products on
8-3 Study Guide page 26 (1-21)