Polynomials

Polynomials are algebraic expressions built from terms like 3x^2, -5x, and 7, and they appear throughout algebra, calculus, physics, and data modeling. Understanding their structure helps students predict how equations behave without graphing every point by hand. The degree, coefficients, and constant term each give clues about the shape and direction of the graph. Factoring adds another powerful tool because it reveals where the graph crosses or touches the x-axis.

A polynomial can be written in standard form, factored form, or sometimes both, and each form highlights different information. The highest exponent gives the degree, which strongly affects end behavior and the possible number of turning points. Factors such as (x - 2) or (x + 1)^2 show zeros and multiplicities, which tell whether the graph crosses the axis or just touches it. By connecting algebraic form to graph behavior, students can move smoothly between equations, tables, and sketches.

Key Facts

A polynomial in standard form looks like a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where n is a whole number.
The degree of a polynomial is the greatest exponent with a nonzero coefficient.
If f(x) = a(x - r)^m, then x = r is a zero with multiplicity m.
Even multiplicity usually means the graph touches the x-axis and turns around at that zero.
Odd multiplicity usually means the graph crosses the x-axis at that zero.
For degree n, a polynomial can have at most n real zeros and at most n - 1 turning points.

Vocabulary

Polynomial: An expression made of terms with variables raised to whole-number exponents and combined by addition or subtraction.
Degree: The highest exponent of the variable in a polynomial after like terms are combined.
Coefficient: The numerical factor multiplying a variable term, such as 4 in 4x^3.
Factor: A simpler expression that multiplies with others to produce the polynomial.
Multiplicity: The number of times a particular factor is repeated, which affects how the graph behaves at that zero.

Common Mistakes to Avoid

Calling the degree the number of terms, which is wrong because degree depends on the highest exponent, not how many terms appear.
Forgetting to combine like terms before finding the degree, which can lead to choosing an exponent from a term that cancels out.
Assuming every zero makes the graph cross the x-axis, which is wrong because zeros with even multiplicity usually make the graph touch and turn.
Ignoring the leading coefficient when predicting end behavior, which is wrong because both the degree and the sign of the leading coefficient control the graph's ends.

Practice Questions

1 Find the degree, leading coefficient, and constant term of P(x) = -3x^4 + 2x^2 - 7x + 5.
2 For f(x) = (x - 1)^2(x + 3), list all zeros, give each multiplicity, and state whether the graph crosses or touches the x-axis at each zero.
3 A polynomial has odd degree and a negative leading coefficient. Describe the end behavior of its graph and explain why.