Algebra I

Operations with Polynomials

Big Picture

Monomials and polynomials can contain numbers, variables, and exponents. They can be added, subtracted, multiplied, divided, and factored, just like real numbers. There are a few special products of polynomials that are important to know, such as the product of two binomials.

Key Terms

Monomial: A number, a variable with a positive integer exponent, or the product of a number and variable(s) with positive integer exponents.

Polynomial: A monomial or sum of monomials

Term: A part of the polynomial that is added or subtracted.

Coefficient: A number that appears in front of a variable.

Constant: A number without a variable.

Binomial: A polynomial with two terms.

Trinomial: A polynomial with three terms.

Standard  Form: A form where the terms in the polynomial are arranged in order of decreasing power (exponents decrease from left to right)

Leading Coefficient: The coefficient of the term with the greatest power.

Degree of a Monomial: Sum of the exponents in the monomial.

Degree of a Polynomial: The greatest degree of the terms.

Like Terms: Terms in the polynomial with the same exponents (coefficients could be different).

Terminology

Examples of monomials:

  • \(7,\frac{1}{2}x, 3a^2b\)

These are not monomials:

  • \(\frac{3}{x},2^a,x^{-1}\)

A polynomial is made up of different terms that contain positive integer powers of the variables.

  • A term can be a coefficient with a variable or just a constant.
  • A polynomial with only two terms is called binomial, and a polynomial with only three terms is called a trinomial.
  • If the terms are written in standard form so that the exponents decreased from left to right, the first coefficient is the leading coefficient.

     

  • \(4\) is the coefficient of \(x^3\) and is the leading coefficient
    Degrees:
  • \(4x^3\) has degree \(3\)
  • \(2x^2\) has degree \(2\)
  • \(-3x\) has degree \(1\)
  • \(1\) has degree \(0\)
  • The degree of the polynomial is \(3\)

    Addition & Subtraction of Polynomials

    • To add \(2\) or more polynomials, write their sum and combine like terms. Once there are no more like terms, thepolynomial is simplified.
    • To subtract \(1\) polynomial from the other, add the opposite of each term of the polynomial we are subtracting.

    Example: \((4x^2-3xy+2) + (2x^3+5y) - (x^2+5xy-3)\)

    Group like terms: \((2x3) + (4x^2-x^2) + (-3xy-5xy) + (5y) + (2-(-3))\)

    Simplify: \(2x^3 + 3x^2 - 8xy + 5y + 5\)

      Algebra I

      Operations with Polynomials cont.

      Multiplication of Polynomials

      Multiplying Monomials

      • Multiply the coefficients as we would any number and use the product rule for exponents.
      • The product rule for exponents is \(x^n ∙ x^m = x^{n+m}.\)

      Multiplying Monomials

      • Use the distributive property so that every term in one polynomial is multiplied by every other term in the other polynomial.
      • The distributive property is \(a(b+c) = ab+ac\).

      Another method is called FOIL. If given \((a+b)(c+d)\):

      • Multiply the First terms in each polynomial \((a, c)\)
      • Multiply the Outermost terms in each polynomial \((a, d)\)
      • Multiply the Innermost terms in each polynomial \((b, c)\)
      • Multiply the Last terms in each polynomial \((b, d)\)
      • Combine any like terms

      So \((a+b)(c+d) = ac + ad + bc + bd\)

      Polynomials can be multiplied vertically, similar to vertical multiplication with regular numbers.

      Example: \((a+b)(c+d)\)

      \(\begin{array}{r}
      &a+b\\
      \times\!\!\!\!\!\!&c+d\\
      \hline
      &ad+bd\\
      +ac+bc\!\!\!\!\!\!&\\
      \hline
      &ac+ad+bc+bd\\
      \end{array}\)

      Symbol
      Meaning
      +
      Add
      -
      Substract
      X or .
      Multiply
      \(\div\) or /
      Divide
      \(\sqrt{}\) or \(\sqrt[n]{}\)
      square root, nth root
      | |
      Absolute value
      =
      Equals
      Not equal
      Approximately equal
      <, ≤
      Less than, less than or equal to
      >, ≥
      Greater than, greater than or equal to
      {  }
      Set symbol
      \(\in\)
      An element of a set
      ( ), [ ]
      Group symbols

      Division of Polynomials

      Dividing Monomials

      • Write as a fraction and use the quotient of powers.
      • The quotient rule for exponents is \(\frac{x^n}{x^m} = x^{n-m}\)

      Dividing Polynomials

      • To divide a polynomial by a monomial, we can divide each term in the numerator by the monomial.
      • Example:
        \(\frac{3x^3 + 6x - 1}{x} = \frac{3x^3}{x} + \frac{6x}{x} - \frac{1}{x} = 3x^2 + 6 - \frac{1}{x}\)
      • To divide a polynomial by a binomial, use long division.
      • Dividend \(\div\) Devisor = Quotient + \(\frac{Remainder}{Divisor}\)
      • The dividend is the numerator, and the divisor isthe denominator.

      For example: \(\frac{x^2 + 4x + 5}{x + 3}\)

      \(\require{enclose}
      \begin{array}{rll}
         x+1 && \\[-3pt]
        x+3 \enclose{longdiv}{x^2 + 4x +5}\kern-.2ex \\[-3pt]
           \underline{-x^2 - 3x\phantom{00}} && \\[-3pt]
           x + 5\phantom{0} && \\[-3pt]
           \underline{\phantom{0}-x - 3\phantom{0}} && \\[-3pt]
           \phantom{0}2 &&  \\[-3pt]
          \end{array}\)

      So \(\frac{x^2 + 4x + 5}{x + 3} = x + 1 + \frac{2}{x + 3}\)

      Tips:

      • Rewrite the polynomial in standard form.
      • Write any missing terms with zero coefficients.
      • Example: Rewrite \(2x^2 + 3\) as \(2x^2 + 0x + 3\)

      Special Products of Polynomials

      Square of a Binomial

      \((a+b)^2 = (a+b)(a+b)
                  = a^2+ab+ab+b^2
                  = a^2+2ab+b^2\)

      You can also remember the square of the binomial by drawing this diagram:

      The area of the square is \((a+b)(a+b) = (a+b)^2\)

      • The area can be found by adding up the four smaller squares and rectangles.
      • \((a+b)(a+b) = \textcolor{#90cde2}a^\textcolor{#90cde2}2+2\textcolor{#e2c802}a\textcolor{#e2c802}b+\textcolor{#df2e24}b^\textcolor{#df2e24}2\)

      We can also find \((a-b)^2\) by replacing \(b\) with \(-b\):

      \((a-b)^2 = (a-b)(a-b)
                 = a^2-ab-ab-b^2
                  = a^2-2ab+b^2\)

      Sum and Difference Patterns

      \((a+b)(a-b) = a^2+ab-ab+b^2
                        = a^2-b^2\)

      \(a\) and \(b\) can represent numbers, variables, or variable expressions.