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Chapter 3 Expressions and Polynomials

Chapter 3 Expressions and Polynomials. 3.1 Evaluating Expressions. Differentiate between an expression and an equation Equation : A mathematical relationship that contains an equal sign Expression : A collection of constants, variables, and Operations

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Chapter 3 Expressions and Polynomials

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  1. Chapter 3 Expressions and Polynomials

  2. 3.1 Evaluating Expressions • Differentiate between an expression and an equation Equation: A mathematical relationship that contains an equal sign Expression : A collection of constants, variables, and Operations F = ma is an equation , whereas ma is an expression 5 + 6 = 11, is an equation, where as 5+6 is an Expression • To evaluate expression • Replace the variables with the corresponding given values • Calculate using the order of operations agreement Division Properties When zero is the divisor with any dividend other than zero, the quotient is Undefined n/0 = undefined when n = 0 0/n = 0, when n = 0 0/0 = indeterminate

  3. 3.2 Introduction to Polynomials • Identify monomials Monomial or Term : An algebraic expression that is a constant, or a product of a constant and variables that are raised to whole –number Powers. x6 , x2y, 5x, -4xy3 , x3 y 3 • Identify the coefficient and degree of a monomial Coefficient : The numerical factor in a monomial Degree of a monomial : The sum of the exponents on all variables in a monomial • Identify the like terms Like terms: Monomials that have the same variables raised to the same exponents x6 + 2 x6 = 3 x6 2x2y + 3 x2y = 5 x2y • 4xy3 + 6xy3 = 2 xy3 x3 y 3 + 3x3 y 3 = 4 x3 y 3 x3 y 3 - 5 x3 y 3 = -4 x3 y 3

  4. Polynomials are algebraic expressions that are similar to whole numbers written in expanded notation For example 2394= 2.103+ 3.102+9.10+4 Expanded form , written with base 10 2x 3+ 3x2 + 9x + 4 Polynomial form , written with base x • Identify the polynomials and their terms Polynomial : A monomial or an expression that can be written as a sum of Monomials Polynomial in one variable: A polynomial with only one variable 2x 3+ 3x2 + 9x + 4 Multivariable Polynomial: A polynomial with more than one variable 2x 3 y+ 3z2 • Identify the degree of a polynomial Degree of a multiple-term polynomial: The greatest degree of all the terms that make up the polynomial 2x3+ 3x2 + 9x + 4 To write a polynomial in descending order of degree, place the term with greatest degree first, then the term with the next greatest degree, and so on. 2x 3+ 3x2 + 9x + 4

  5. 3.3 Simplifying the polynomials To combine the like term • Add or subtract the coefficients • Keep the variables and their exponents the same Examples 3x + 8x + 9x = 20x 2x2 + 5x2 = 7x2 - 8y3 + 6y3 = -2y3 3x2 – 5 + 4x 3 + 8x + 7x 2- 3x + 9 Combine like terms 4x 3 +7x 2 + 3x2 + 8x – 3x + 9 – 5 = 4x 3 + 10x 2 + 5x + 4

  6. 3.4 Adding and Subtracting Polynomials • Add polynomials in one variable (2x 3+ 3x2 - x2 + 9x + 1) + (4x2 - 7x + 4 ) = 2x 3 + 3x2 + 4x2 - 9x – 7x + 1 + 4 = 2x 3 + 7x2 – 16x + 5 • Write an expression for the perimeter of a given shape Perimeter = 2( Length + width) = 2( 2x + 1 + 3x + 2) = 2( 2x + 3x + 1 + 2) ( combine like term) = 2(5x + 3) = 10x + 6 • Subtract polynomials in one variable 7x2 + 10x + 5 – (2x2 + 6x + 5 ) = 7x2 - 2x2 + 10x - 6x + 5 – 5 = 5x2 + 4x + 0 3x + 2 2x + 1

  7. 3.5 Multiplying Polynomials Rule – When multiplying exponential forms that have he same base, we can add the exponent and keep the same base 2 2 . 23 = 4. 8 = 32 2.2. 2.2.2 25 = 32 Alternative way 22 . 23 = 22+3 = 25 (x3 ) (x4 ) x3 means 3x’s, x4 means 4x’s =( x.x.x)(x.x.x.x) Total 7x’s = x 7

  8. RulesMultiply monomials 1. When multiplying exponential forms that have the same base, we can add the exponents and keep the same base n a nb = na +b Simplify monomials raised to a power • To simplify an exponential form raised to a power, we can multiply the exponents and keep the same base ( n a ) b = nab Multiply polynomials • To multiply a polynomial by a monomial, use the distributive property to multiply each term in the polynomial by the monomial 3x( 2x2 + 3x + 1) 6x3 9x2 3x Multiply two polynomials: 4. Multiply every term in the first polynomial by every term in the second polynomial Combine like terms (x + 6) (x + 7) = x2 + 6x + 7x + 42 = x2 + 13x + 42 • The product of two conjugates is a difference of two squares2x (a + b)(a - b) = a2 - b2

  9. 3.6 Prime Numbers and GCF Determine if a number is prime, composite, or neither Prime number: A natural number other than 1 that has exactly two different factors List of primes : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47……. Composite Number: A natural number that has factors other than 1 and itself 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25 0 and 1 neither prime nor composite To determine if a given number is prime or composite, divide the given number by the primes on the list of prime numbers and consider the results

  10. Prime Factorization To find the prime factorization of a number, use a factor Tree • Draw two branches below the number • Place two factors that multiply to equal the given number at the end of the two branches • Repeat steps 1 and 2 for every composite factor. • Place all the prime factors together in a multiplication sentence 84 84 84 84 2 42 4 21 7 12 3 28 2 21 22 3 7 2 6 2 14 3 7 2 3 2 7

  11. Find all possible factors of a given numberFind the greatest common factor of a given set of numbers using prime factorization. 24 = 2.2.2.3 = 23 .3 60= 2.2.3.5 = 22.3.5 GCF = 2.2.3 = 22 .3 = 12 Set of monomials 12x4 and 9x3 12x4 = 22 .3.x4 9x3 = 32 . X3 GCF = 3x3

  12. 3.7 Introduction to factoring • Divide monomials Rules When dividing exponentials forms that have the same base, we can subtract the divisor’s exponents from the dividend’s exponent and keep the same base. na/nb= n a – b , where n = b Example x4 / x2 =x4-2=x2 4x4 / 2x3 = 2 x4-3= 2x But 0a /0b= Indeterminate Any base other than 0 power simplifies to the number 1 n0 = 1, when n = 0 00 = Indeterminate

  13. Procedure To divide monomials: • Divide the coefficients. 3. For like bases, subtract the exponent of the divisor base from the exponent of the dividend base and keep the base. If the bases have the same exponent, then they divide out, becoming 1. 4. Bases in the dividend that have no like base in the divisor are written unchanged in the quotient. • To divide a polynomial by a monomial. • Divide each term in the polynomial dividend by the monomial divisor • Simplify

  14. Factor the GCF out of a polynomial Procedure To factor a monomial GCF out of a given polynomial • Find the GCF of the terms that make up the polynomial • Rewrite the polynomial as a product of the GCF and parentheses that contain the quotient of the given polynomial and the GCF. Given polynomial = GCF Given polynomial GCF 3. Simplify in the parentheses

  15. ExamplesDivide • 16x4 - 8x3 + 4x2 4x2 = 16x4 - 8x3 + 4x2 (Divide each term by 4x2 ) 4x2 4x2 4x2 = 4x2 - 2x + 1 • Factor 18y – 6 The GCF of 18 and 6 = 6 = 6(18y – 6) 6 = 6 ( 18y - 6) ( Divide each term of the polynomial by the GCF ) 6 6 = 6(3y – 1) ( Simplify the paranthesis) Check 6(3y – 1) ( Distributive property) = 18y - 6

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