Kid-friendly explanation
A polynomial is a math expression built from pieces like x, x2, x3, and numbers, all added or subtracted. Each piece is called a term. You will not see variables in the denominator or square roots of x in a polynomial. It is like a stack of simple power pieces put together.
Worked examples
Example 1 (middle school level): Is this a polynomial?
Look at the expression 3x2 - 4x + 7.
Each term is a number times a power of x with a whole-number exponent: 3x2, -4x (which is -4x1), and 7 (which is 7x0). There are no variables in denominators, no negative exponents, and no square roots of x. So 3x2 - 4x + 7 is a polynomial.
It has degree 2, because the highest exponent on x is 2. It has 3 terms, so it is called a trinomial.
Example 2 (high school level): Using zeros to sketch a polynomial function
Suppose a function is given by f(x) = (x - 1)2(x + 2).
First, identify its zeros. A zero is a value of x that makes f(x) = 0. From (x - 1)2(x + 2) = 0, the zeros are x = 1 and x = -2.
The factor (x - 1)2 has exponent 2, so the zero x = 1 has multiplicity 2. The factor (x + 2) has exponent 1, so the zero x = -2 has multiplicity 1.
Multiplicity 1 means the graph crosses the x-axis at that zero. So the graph crosses at x = -2. Multiplicity 2 means the graph touches the x-axis and turns around at that zero. So the graph touches and bounces at x = 1.
To see the end behavior, expand just enough to find the leading term. Multiply the leading parts: x2 from (x - 1)2 and x from (x + 2). The leading term is x3, which has a positive coefficient and odd degree 3.
For a polynomial with leading term x3, as x goes to the right (x → ∞), f(x) goes up (f(x) → ∞). As x goes to the left (x → -∞), f(x) goes down (f(x) → -∞). Using the zeros, multiplicities, and end behavior, your child can sketch a reasonable graph of f(x).
Going deeper with polynomials
In later grades, polynomials are treated as a special type of function when written like f(x) = 2x3 - x + 5. Your child learns to recognize a polynomial function from its equation, a table of values, or a graph, and to tell it apart from an exponential function or a rational function.
Operations on polynomials are important. Polynomials are closed under addition, subtraction, and multiplication, which means that if you add, subtract, or multiply two polynomials, the result is still a polynomial. Students learn to add, subtract, and multiply polynomials, then move on to dividing polynomials using long division or synthetic division.
Factoring and polynomial identities connect polynomials to equations and graphs. Factoring rewrites a polynomial as a product of simpler polynomials, which reveals its zeros and x-intercepts. Polynomial identities, such as (a + b)2 = a2 + 2ab + b2, describe patterns that are true for all values of the variables and are used to simplify expressions and solve equations.
Common mistakes with polynomials
- Thinking any expression with x is a polynomial. For example, 3/x + 2 has a variable in the denominator, so it is not a polynomial. A polynomial only has variables in the numerator with whole-number exponents.
- Misclassifying the degree of a polynomial. A student might say 4x3 - 2x5 + 7 has degree 3 because 3 is the first exponent seen. The degree is actually 5, the largest exponent on the variable.
- Adding or subtracting unlike terms incorrectly. A student might simplify 3x2 + 5x - x2 as 3x2 + 5x - x, getting 2x2 + 4x, which is wrong. The correct combination is (3x2 - x2) + 5x = 2x2 + 5x, because only like terms with the same exponent can be combined.
How polynomials show up in everyday life
Polynomials model many real situations, even if the word itself is not used. The height of a ball after it is thrown, the shape of a bridge arch, or the total cost of buying several items with a discount can all be described with polynomial functions. When your child studies motion, area, or profit problems in algebra, there is often a polynomial behind the scenes.