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5.1 – Polynomial Basics

Key Terms

  • Power – A small raised number that tells you how many times to multiply the base number by itself.
    • It is also called an exponent.
    • Ex. 2^5=2 \cdot 2\cdot 2\cdot 2\cdot 2
    • Degree is the exponent value
      • Highest degree is the highest value of all of the exponents in the polynomial.
  • Common Factors – Numbers or expressions that are factors of two or more numbers or polynomials.
  • Polynomial – An algebraic expression with one or more terms.
    • None of the variables are in the denominator of a fraction, and any exponents are whole numbers.
      • Ex. \frac{2}{3}y^3-y^2+15 is a polynomial.
      • Ex. 3x^2+\frac{5}{x}-2 is a NOT a polynomial because there is a variable (x) in the denominator of the fraction.

 

Review

  • Nomial – Term
    • Terms are separated by plus and minus signs.
    • Monomial – 1 term
      • Ex. 8x^2
      • Ex. 4
      • Ex. y
      • Ex. p^4
    • Binomial – 2 terms
      • Ex. 8x^2+1
      • Ex. x-1
    • Trinomial – 3 terms
      • Ex. 8x^2+3x-1
      • Ex. 4a^4-10x+16

 

  • Coefficient – A number that is multiplied by a variable.
    • Ex. 24x^8: the 24 is the coefficient
    • Ex. -5x: the -5 is the coefficient

 

  • Variables with Negative Exponents Become Fractions with Variables in the Denominator: n^{-a}=\frac{1}{n^a}
    • These are NOT monomials, and not part of polynomials
    • Ex. 3x^{-4} is NOT a monomial

 

  • Square Roots of Variables Become Variables to Fractional Exponents: \sqrt{x}=x^{\frac{1}{2}}
    • These are NOT monomials, and not part of polynomials
    • Ex. 2 \sqrt{x} is NOT a monomial

 

  • Polynomials in Standard Form are Written in Descending Order
    • Ex. 5x^5+13x^4-x^3-2x^2+6x-10

 

  • Constants – A number times x to the zero power
    • Ex: 4=4x^0
    • Ex: -15=-15a^0
      • Use whatever variable is being used in the rest of the polynomial

Alg2B 5.1 - Polynomial Rule

 

 


Notes

  • Polynomial Conditions – ALL of the conditions below MUST be met for the algebraic expression to be a polynomial:
    • Not all algebraic expressions are polynomials.
    • An expression is a polynomial if and only if it meets all of the following conditions:
      • There are no fractions that contain variables in the denominator.
      • There are only whole number exponents (0, 1, 2, …) — no fractional or negative exponents.
      • There are no radical expressions that contain variables.
      • All coefficients are real, though they may be fractions, radicals, or irrational numbers.

 

  • Closed Systems and Polynomials
    • Closed System – Adding, subtracting, or multiplying polynomials will result in another polynomial.
    • Not a Closed System – Dividing polynomials may or may not result in a polynomial.
      • If the quotient has a variable in the denominator of a fraction, it is NOT a Closed System.

 


  • Adding Polynomials
    • Combine like terms
    • Put the terms in descending order (from greatest to smallest exponent values)

 


  • Subtracting Polynomials
    • Distribute the -1 before you combine like terms.

Alg2B 5.1 - Subtraction

 


  • Multiplying Binomials Using FOIL

Alg2B 5.1 - FOIL

 

  • FOIL: Special Cases

Alg2B 5.1 - Special Case 1 Alg2B 5.1 - Special Case 2 Alg2B 5.1 - Special Case 3

 

  • Multiplying Polynomials
    • Distribute the smaller polynomial over the larger polynomial.
    • Example:

Alg2B 5.1 - Multiply Polyn

 

    • Multiplying Polynomials Vertically

 

    • Multiplying Trinomials

Alg2B Mult Trinomials

 


  • Dividing Polynomials
    • It is possible to have a remainder when dividing polynomials.
    • The degree of the remainder will always be less than the degree of the divisor.

Alg2B 5.1 - Division
Alg2B 5.1 - Division Ex

 

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