POLYNOMIALS€¦ · MULTIPLYING A BINOMIAL BY A BINOMIAL HORIZONTAL (FOIL) When 2 binomials are...

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UNIT 6 POLYNOMIALS

Transcript of POLYNOMIALS€¦ · MULTIPLYING A BINOMIAL BY A BINOMIAL HORIZONTAL (FOIL) When 2 binomials are...

Page 1: POLYNOMIALS€¦ · 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

UNIT 6POLYNOMIALS

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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

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Let's Practice Finding the Degree of a Monomial

3 degree of ___

3x degree of ___

3x2 degree of ___

3x3 degree of ___

3x4 degree of ___

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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.

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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.

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Use this chart as a guide

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Let's Practice (Fill out this table)

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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.

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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.

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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

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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)

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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)

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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)

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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)

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