The Main Idea 

Polynomial functions are like the DNA of algebra—they combine simplicity and complexity to form an incredibly diverse array of functions.

At their core, polynomials are sums of terms made up of coefficients and variables raised to whole number powers. The degree of a polynomial, given by the highest power of the variable, tells us a lot about the function’s behavior and the shape of its graph.

Let [latex]n[/latex]  be a non-negative integer. A polynomial function is a function that can be written in the form

[latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]

This is called the general form of a polynomial function.

Each [latex]{a}_{i}[/latex] is a coefficient and can be any real number.

Each product [latex]{a}_{i}{x}^{i}[/latex] is a term of a polynomial function.

Quick Tips

• Linear, Quadratic, and Beyond: A first-degree polynomial is linear, a second-degree is quadratic, and higher degrees have their own characteristics and complexities.
• Coefficient Clues: The coefficients in a polynomial can tell us about the steepness and direction of the graph. A positive leading coefficient means the graph opens upward, and a negative one indicates it opens downward.

The Main Idea 

The degree of a polynomial is the highest power of the variable present. The leading coefficient is the coefficient of the term with the highest power. These two characteristics can tell us a lot about the function’s behavior, especially its growth and end behavior.

• Degree Tells the Tale: The degree of a polynomial function hints at the number of roots and the possible number of turns on its graph. For instance, a second-degree polynomial, or a quadratic, will have at most two roots and one turn.
• Leading Coefficient Impact: The leading coefficient isn’t just a number—it’s the boss. It influences the end behavior of the polynomial’s graph. If it’s positive, the graph eventually rises; if negative, the graph falls.

Quick Tips: Identifying Degree and Leading Coefficient

• Arrange terms in descending order of power to easily identify the leading term and coefficient.
• The degree gives a sense of the ‘shape’ of the graph of the polynomial function.
• For instance, [latex]f(x)=3+2x^2−4x^3[/latex] has a degree of [latex]3[/latex] and a leading coefficient of [latex]-4[/latex], indicating a cubic function with a negative leading term.

The Main Idea

• Like Terms:
  • Expressions with the same variables raised to the same powers
  • Only coefficients are combined; exponents remain unchanged
• Adding Monomials:
  • Combine coefficients of like terms
  • Keep the variable and its exponent the same
• Subtracting Monomials:
  • Similar to addition, but pay attention to signs
  • Remember that subtracting a negative is the same as adding a positive
• Adding Polynomials:
  • Combine like terms across all polynomials
  • Use the Commutative Property to rearrange terms if needed
• Subtracting Polynomials:
  • Distribute the negative sign to all terms in the subtracted polynomial
  • Then proceed as with addition

The Main Idea

• Distributive Property:
  • Key principle for multiplying polynomials
  • [latex]a(b + c) = ab + ac[/latex]
• General Polynomial Multiplication:
  • Multiply each term of the first polynomial by every term of the second
  • Combine like terms in the result
• FOIL Method:
  • Specific technique for multiplying two binomials
  • FOIL stands for First, Outer, Inner, Last

The Main Idea

• Perfect Square Trinomials:
  • Result from squaring a binomial
  • General form: [latex](a + b)^2 = a^2 + 2ab + b^2[/latex]
  • Also applies to subtraction: [latex](a - b)^2 = a^2 - 2ab + b^2[/latex]
• Characteristics of Perfect Square Trinomials:
  • First and last terms are perfect squares
  • Middle term is twice the product of the terms in the original binomial
  • Sign of the middle term matches the sign in the original binomial
• Difference of Squares:
  • Result from multiplying sum and difference of the same terms
  • General form: [latex](a + b)(a - b) = a^2 - b^2[/latex]
  • Middle terms cancel out
• No Special Form for Sum of Squares:
  • [latex](a + b)^2 \neq a^2 + b^2[/latex]
  • Always results in a perfect square trinomial

The Main Idea

• Extension of Single-Variable Operations:
  • Rules for polynomial operations apply to multi-variable polynomials
  • Operations include addition, subtraction, and multiplication
• Like Terms in Multi-Variable Polynomials:
  • Terms with the same variables raised to the same powers
  • Coefficients of like terms can be combined
• Distributive Property:
  • Key principle in multiplying multi-variable polynomials
  • Applied similarly to single-variable polynomials
• Combining Like Terms:
  • This is the essential step after applying the distributive property
  • Terms with different variables or exponents are not combined