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8-1: adding and subtracting polynomials, Study notes of Pre-Calculus

University of California - BerkeleyPre-Calculus

EXAMPLE 1: DEGREE OF A MONOMIAL. Find the degree of each monomial. 1. 2. 18. 3. 4. EXAMPLE 2: CLASSIFYING POLYNOMIALS. A polynomial is a monomial or the sum ...

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8-1:%%ADDING%AND%SUBTRACTING%POLYNOMIALS%
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Lesson&Objectives:&
โ€ข Classify&polynomials&
โ€ข Add&and&subtract&polynomials&
A!monomial!is!an!expression!that!is!a!number,!a!variable,!or!a!product!of!a!number!and!one!or!more!
variables.!
!
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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.!
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EXAMPLE+1:++DEGREE+OF+A+MONOMIAL!
Find!the!degree!of!each!monomial.!
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1.!! ! ! ! 2.!!18! ! ! ! 3.!! !!!!4.!! !
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EXAMPLE+2:!!CLASSIFYING+POLYNOMIALS!
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A!polynomial!is!a!monomial!or!the!sum!or!difference!of!two!or!more!monomials.!
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The!polynomial!shown!above!is!in!standard!form.!!Standard+form+of+a+polynomial!means!that!the!
degrees!of!its!monomial!terms!decrease!from!left!to!right.!
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The!degree+of+a+polynomial!in!one!variable!is!the!same!as!the!degree!of!the!monomial!with!the!greatest!
exponent.!!The!degree!of! !is!4.!
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After!you!simplify!a!polynomial!by!combining!like!terms,!you!can!name!the!polynomial!based!on!its!
degree!of!the!number!of!monomials!it!contains.!
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Polynomial+
Degree+
Name+Using+
Degree+
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3xy3
5m
4x5
3x4+5x2โˆ’7x+1
3x4+5x2โˆ’7x+1
7x+4
3x2+2x+1
4x3
3x4+5x2โˆ’7x+1
pf3
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Download 8-1: adding and subtracting polynomials and more Study notes Pre-Calculus in PDF only on Docsity!

8 - 1: ADDING AND SUBTRACTING POLYNOMIALS

Lesson Objectives:

  • Classify polynomials
  • Add and subtract polynomials A monomial is an expression that is a number, a variable, or a product of a number and one or more variables. 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. EXAMPLE 1: DEGREE OF A MONOMIAL Find the degree of each monomial.
    1. 18 3. 4. EXAMPLE 2 : CLASSIFYING POLYNOMIALS A polynomial is a monomial or the sum or difference of two or more monomials. The polynomial shown above is in standard form. Standard form of a polynomial means that the degrees of its monomial terms decrease from left to right. The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest exponent. The degree of is 4. After you simplify a polynomial by combining like terms, you can name the polynomial based on its degree of the number of monomials it contains. Polynomial Degree Name Using Degree Number of Terms Name Using Number of Terms 5 3 xy^3 5 m 4 x^5 3 x^4 + 5 x^2 โˆ’ 7 x + 1 3 x 4
  • 5 x 2 โˆ’ 7 x + 1 7 x + 4 3 x^2 + 2 x + 1 4 x 3 9 x^4 + 11 x 3 x 4
  • 5 x 2 โˆ’ 7 x + 1

Write each polynomial in standard form. Then name each polynomial by its degree and the number of its terms.

EXAMPLE 3: ADDING POLYNOMIALS Simplify.

EXAMPLE 4: SUBTRACTING POLYNOMIALS Simplify.

    โˆ’ 2 + 7 x 3 x^5 โˆ’ 2 โˆ’ 2 x^5 + 7 x 6 x^2 + 7 โˆ’ 9 x^4 3 y โˆ’ 4 โˆ’ y^3 8 + 7 v โˆ’ 11 v

( 6 x^2 +^3 x^ +^7 ) +^ (^2 x^2 โˆ’^6 x^ โˆ’^4 ) (^1 2 m^2 +^4 ) +^ (^8 m^2 +^5 ) ( t^2 โˆ’^6 ) +^ (^3 t^2 +^11 )

9 w 3

  • 8 w 2

( ) +^7 w

3

( +^4 ) 2 p

3 โˆ’ 6 p 2

( +^10 p ) +^ โˆ’^9 p

3 โˆ’ 11 p 2

( +^3 p )

( 2 x^3 +^4 x^2 =^6 ) โˆ’^ (^5 x^3 +^2 x^ โˆ’^2 ) ( v^3 +^6 v^2 โˆ’^ v ) โˆ’^ (^9 v^3 โˆ’^7 v^2 +^3 v )

Name __________________________________________ 8 - 1 Practice Worksheet Period ________________ Find the degree of each monomial.

Name each polynomial based on its degree and number of terms.

Write each polynomial in standard form. Then name each polynomial based on its degree and number of terms.

Simplify.

  5. 6 y^2 w^8 7 c^3 6 8 ab^3 5 x 2 โˆ’ 2 x + 3 7 a 3 + 4 a โˆ’ 12

x

  • 5 w 2
  • 2 4 x โˆ’ 3 x^2 c^2 โˆ’ 2 + 4 c 9 z^2 โˆ’ 11 z^2 + 5 z โˆ’ 5 โˆ’ 10 + 4 q^2 โˆ’ 8 q + 3 q^2 ( w 2
  • w โˆ’ 4 ) + ( 7 w 2 โˆ’ 4 w + 8 ) ( g 4
  • 4 g ) + ( 9 g 4
  • 7 g ) ( 7 y 3 โˆ’ 3 y 2
  • 4 y ) + ( 8 y 4
  • 3 y 2 ) ( 7 h^2 + 4 h โˆ’ 8 ) โˆ’ ( 3 h^2 โˆ’ 2 h + 10 ) ( 24 x^5 + 12 x ) โˆ’ ( x^3 + 8 x^2 โˆ’ x ) (โˆ’ 5 x^4 + x^2 ) โˆ’ (โˆ’ x^3 + 8 x^2 โˆ’ x ) ( 2 x^3 โˆ’ 5 x^2 โˆ’ 1 ) โˆ’ ( 8 x^3 + 3 โˆ’ 8 x^2 ) ( 7 a^3 โˆ’ a + 3 a^2 ) + ( 8 a^2 โˆ’ 3 a โˆ’ 4 ) ( 6 g โˆ’ 7 g^8 ) โˆ’ ( 4 g + 2 g^3 + 11 g^2 ) (โˆ’ 4 t 4 โˆ’ 9 t + 6 ) + ( 13 t + 5 t 4 ) ( 11 + k 3 โˆ’ 6 k 4 ) โˆ’ ( k 2 โˆ’ k 4 ) ( ab 2
  • ba 3 ) + ( 4 a 3 b โˆ’ ab 2 โˆ’ 5 ab ) ( 9 pq 6 โˆ’ 11 p 4 q ) โˆ’ (โˆ’ 5 pq 6
  • p 4 q 4 )

Find an expression for the perimeter of each figure.

Find an expression for each missing length.

  1. Perimeter = 25x + 8 29. Perimeter = 23a โ€“ 7
  2. You can model the number of men and women in the United States who enrolled in college within a year of graduating from high school with the linear equations shown below. Let t equal the year of enrollment, with t = 0 corresponding to 1990. Let m(t) equal the number of men in thousands, and let w(t) equal the number of women in thousands. m(t) = 35.4t + 1146.8 men enrolled in college w(t) = 21.6t + 1185.5 women enrolled in college a) Add the expressions on the right side of each equation to model the total number of recent high school graduates p(t) who enrolled in college between 1990 and 1998. b) Use the equation you created in part a) to find the number of high school graduates who enrolled in college in 1995. c) If you had subtracted the expression on the right side of each equation above, what information would the resulting expression model?