1 algebra 1 adding and subtracting polynomials mr. j. grossman

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ALGEBRA 1ALGEBRA 1

Adding and Subtracting Adding and Subtracting PolynomialsPolynomials

Mr. J. Grossman

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VARIABLE EXPRESSIONSVARIABLE EXPRESSIONS

A variable expression contains one or A variable expression contains one or more variables.more variables.

There is not an equal (=) sign.There is not an equal (=) sign.

25 4 4x xy y

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TERMSTERMS

The terms are the variables, numbers or The terms are the variables, numbers or combinations of variables and numbers combinations of variables and numbers that make up the expression.that make up the expression. There are 4 terms in the given There are 4 terms in the given

expression.expression. Variable terms contain variables.Variable terms contain variables.

There are 3 variable terms in the given There are 3 variable terms in the given expression. What are they?expression. What are they?

Constant terms contain only numbers.Constant terms contain only numbers. There is 1 constant term in the given There is 1 constant term in the given

expression. What is it?expression. What is it?

25 4 4x xy y

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NUMERICAL NUMERICAL COEFFICIENTCOEFFICIENT

Each term contains two parts:Each term contains two parts:

1.1. The variable part and The variable part and

2.2. The numerical coefficientThe numerical coefficient The numerical coefficient is often called The numerical coefficient is often called

simply the coefficient.simply the coefficient. It is the number that multiplies (sits in front It is the number that multiplies (sits in front

of) the variable(s).of) the variable(s). There is always a number in front of a There is always a number in front of a

variable.variable. If you don’t see a number, then the number If you don’t see a number, then the number

is is 11..

25 4 4x xy y

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LIKE TERMSLIKE TERMS

Terms that are identical in the variable Terms that are identical in the variable part (to include exponent value) are part (to include exponent value) are like like termsterms.. Note: Like signs are different from like Note: Like signs are different from like

terms.terms. Constant terms are like terms because Constant terms are like terms because

they have no variable part and are they have no variable part and are therefore alike in the variable part.therefore alike in the variable part.

3 3 25 2 6 2 2 9x y x y

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COMBINING LIKE TERMSCOMBINING LIKE TERMS

Combine or Combine or simplify like simplify like terms by terms by adding or adding or subtracting subtracting their numerical their numerical coefficients.coefficients.

Rearrange the Rearrange the terms if terms if necessary. necessary.

Use the sign Use the sign rules correctly. rules correctly.

3 2 5

3 5 2

2 2

a b a

a a b

a b

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Classifying PolynomialsClassifying Polynomials

Monomial: an expression that is a Monomial: an expression that is a number, a variable, or the product number, a variable, or the product of a number and one or more of a number and one or more variables. variables. a number: 8, a number: 8, a variable: x, or a variable: x, or a product of numbers and variables: 8xya product of numbers and variables: 8xy

Binomial: Binomial: a polynomial with two terms: 2x + 3a polynomial with two terms: 2x + 3

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TrinomialTrinomial

a polynomial with three termsa polynomial with three terms

PolynomialPolynomial

a variable expression in which a variable expression in which the terms are monomials: the terms are monomials:

5x + 9y – 10 + 6xy5x + 9y – 10 + 6xy

3 22 5 7x x

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Standard Form of a PolynomialStandard Form of a Polynomial Written in descending orderWritten in descending order

Arranging the terms of a polynomial, Arranging the terms of a polynomial, according to the exponents, in order according to the exponents, in order from largest to smallest.from largest to smallest.

Degree of a PolynomialDegree of a Polynomial The degree of the term with the largest The degree of the term with the largest

degreedegree

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Constant Constant – a polynomial of 0 degree– a polynomial of 0 degree LinearLinear – a polynomial of 1 degree – a polynomial of 1 degree QuadraticQuadratic – a polynomial of 2 – a polynomial of 2

degreedegree CubicCubic – a polynomial of 3 degree – a polynomial of 3 degree

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PolynomialPolynomial DegreeDegree Name Using Name Using DegreeDegree

Nr of Nr of TermsTerms

Name Using Name Using Nr of TermsNr of Terms

11 LinearLinear 22 BinomialBinomial

22 QuadraticQuadratic 33 TrinomialTrinomial

33 CubicCubic 11 MonomiaMonomiall

00 ConstantConstant 11 MonomiaMonomiall

47 x

123 2 xx34x

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ADDITION & SUBTRACTION ADDITION & SUBTRACTION OF POLYNOMIALSOF POLYNOMIALS

Add or subtract polynomials by Add or subtract polynomials by combining like terms.combining like terms. Add or subtract their coefficientsAdd or subtract their coefficients

2 3

2

3

3 2

vertical format:

3 7 8 3 5 14

3 7 8

3 5 14

3 3 2 6

x x x x

x x

x x

x x x

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Add or subtract polynomials by Add or subtract polynomials by combining like terms.combining like terms. Add or subtract their coefficientsAdd or subtract their coefficients

3 2

3 2

3 2

horizontal format:

2 6 3 7 3 9

2 7 6 3 3 9

2 7 3 6

x x x x

x x x x

x x x

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The minus sign touching the The minus sign touching the parenthesis of the second polynomial parenthesis of the second polynomial means that a negative one is multiplied means that a negative one is multiplied by the second polynomial.by the second polynomial.

Perform multiplication before addition.Perform multiplication before addition. 3 3 2

3 3 2

3 3 2

3 2

6 4 5 2 7 6 2

6 4 5 2 7 6 2

6 2 7 4 6 5 2

8 7 10 7

x x x x x

x x x x x

x x x x x

x x x

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Remove each polynomial from Remove each polynomial from parenthesis, then addparenthesis, then add

3 2

3

2

3 2

13 6 7 5 2 9

13 6 7

5 2 9

13 5 8 2

z z z z

z z

z z

z z z

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Polynomials in more than one Polynomials in more than one variable are added and subtracted by variable are added and subtracted by combining like terms.combining like terms.

3 2 2 3 2 2

3 2 2 3 2 2

3 2 2

5 2 4 2 5

5 2 4 2 5

7 3 5

x y x y xy x y x y xy

x y x y xy x y x y xy

x y x y xy

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Any Any questions??questions??

??

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Let’s PracticeLet’s PracticeText pg 459Text pg 459

Problems 1, 9, 15, Problems 1, 9, 15, 21, 28, 35, 39, 40.21, 28, 35, 39, 40.

1 algebra 1 adding and subtracting polynomials mr. j. grossman

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