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10.1 – What is a Polynomial?

Key Terms

  • Degree – The value of the greatest exponent in a polynomial.
  • Evaluate – To find the numerical value of an expression.
  • Factor – A number or expression that can be multiplied by another number or expression to create a certain product.
  • Factoring – Writing a number or expression as a product of two or more numbers or expressions.
  • Monomial – A polynomial with only one term.
  • 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.

Review

Parts of Polynomials
  • Degree: exponents, powers
  • Coefficients: numbers that are before (and multiplied with) a variable.
    • Ex. 3x: the coefficient is 3
    • Ex. -5a^{2}: the coefficient is -5
  • Constants: numbers without variables
  • Nonnegative numbers: positive numbers or zero
  • Terms: monomials separated by plus and minus signs
 Integers
  • Integers: Negative, Positive and Zero numbers
    • \frac{1}{2} is not an integer
    • \sqrt{x}=x^{\frac{1}{2}} is a root raised to a non-integer power
    • -1 is a negative integer
    • \frac{x}{x-1}=x(x-1)^{-1} is a fraction with a negative exponent

Notes

Monomials
  • A monomial term always has the form ax^{n}, where a is any number and n is a nonnegative integer.
  • The number a is called the coefficient of the term, and n is called the degree of the term.

Alg1B10.1 Monomial Chart Alg1B10.1 Monomial Details

Not Monomials
  • Fractions with variables in the denominator
  • Variables with negative exponents
  • Radicals (√) with variables under the root

 

Polynomials
  • One or more monomials separated by plus and minus signs
    • Plus signs are positive numbers and minus signs are negative numbers
    • Some polynomials have special names
      • Monomials have 1 term (mono means 1)
      • Binomials have 2 terms (bi means 2)
      • Trinomials have 3 terms (tri means 3)
  • Ex 1. -4x^{3}-2x^{2}+6x^{1}-8x^{0} simplifies to -4x^{3}-2x^{2}+6x-8
    • Notice that the x-term is to the 1st power, but you don’t have to write it.
    • Notice that the constant term is to the zero power, but you don’t have to write it.
      • Anything to the zero power equals 1; so, 8 times x^{0} is the same as 8 times 1.
      • 8 times 1 equals 8, so that’s why it is a constant term.
  • Ex 2. \frac{3}{5}x^{2}-2 is a polynomial
    • Remember, the coefficient can be a fraction, but the exponent cannot

 

Degrees and Exponents
  • The greatest exponent in a polynomial is the degree of that polynomial.
    • Ex. -5x^{4}+2x^{3}-\frac{1}{3}x^{2}-18x+3 has a degree of 4
    • Ex. 9x^{18}+4x^{4}-55x--91 has a degree of 18
    • Ex. 3x+5 has a degree of 1 since x is the same as x to the 1st power
    • Ex. 99 has a degree of zero because it is the same as 99x^{0} since x^{0}=1 and 99 times 1 = 99.

Alg1B10.1 Degree Zero

Descending Order
  • You will need to rewrite polynomials in descending order.
    • When a polynomial is written in descending order, its first term is called its leading term.
      • The leading term has a leading coefficient.
    • This is the order from the highest degree to the smallest degree (usually the constant term).
    • How to put polynomials into descending order:

Alg1B10.1 Degree Descending Alg1B10.1 Degree Descending1

 

Evaluating Polynomials
  • Steps to evaluate a polynomial
    1. Substitute a value for the variable
    2. Simplify
  • Ex. Evaluate x = 2 in the polynomial: 3x^{3}-x^{2}+4x-6
    • 3(2)^{3}-(2)^{2}+4(2)-6
    • 3(8)-4+8-6
    • 24-4+8-6
    • 20+8-6
    • 28-6
    • Answer: 22

 

Factoring Polynomials
  • Remember
    • Factors are multiplied
    • Terms are added

Alg1B10.1 Factoring

 

Closed Sets
  • A set has closure under an operation if the performance of that operation on any members of the set always produces a member of the same set.
    • Notice that division is NOT a closed set.

Alg1B10.1 Closed Sets

 

Examples
  • Ex 1. The polynomial 6x^{2}+9x has factors of 3x and what?
    • 3x times what will give you 6x^{2}? Answer: 2x
    • 3x times what will give you 9x? Answer: 3
    • Combined answer: 2x+3
  • Ex 2. Under which of the following operations are the polynomials 11x+3 and 5y-8 not closed?
    • Answer: Division
  • Ex 3. What is the coefficient of the term of degree 3 in this polynomial?  x^{8}+2x^{4}-4x^{3}+x^{2}-1
    • Answer: -4

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