Polynomials: building expressions from powers of x

You already know how to work with a single number, and with a single power like \( x^2 \). A polynomial is simply what you get when you let those pieces team up — a tidy stack of powers of \( x \), each scaled by a number, all added together. Almost every curve you will ever meet in algebra, from the arc of a thrown ball to the shape of a satellite dish, is hiding a polynomial inside it. Learn to read and rearrange them, and a huge part of mathematics suddenly speaks your language.

What a polynomial actually is

Look at this expression:

It is built from three terms, each separated by a plus or minus sign: \( 3x^2 \), then \( 2x \), then \( -5 \). Every term is a number multiplied by a power of \( x \). That number out front is the coefficient. So the coefficient of \( x^2 \) here is \( 3 \), and the coefficient of \( x \) is \( 2 \).

What about the \( -5 \)? It has no \( x \) at all — it is a plain number, called the constant term. You can think of it as \( -5x^0 \), because \( x^0 = 1 \); the power of \( x \) is just zero. That little trick is worth holding onto: every term is a coefficient times a power of \( x \), even the lonely constant.

A polynomial, then, is any finite sum of terms like these. The only rule is that the powers of \( x \) must be whole numbers — \( 0, 1, 2, 3, \dots \) — never fractions or negatives. So \( 4x^3 - x + 7 \) is a polynomial, but \( \frac{2}{x} \) and \( \sqrt{x} \) are not.

Degree, leading term, and the names

The degree of a polynomial is the highest power of \( x \) that appears. In \( 3x^2 + 2x - 5 \), the powers are \( 2 \), \( 1 \), and \( 0 \), so the degree is \( 2 \). The degree is the single most important fact about a polynomial — it controls how the curve behaves and how many times it can cross the \( x \)-axis.

The term carrying that highest power is the leading term, and its coefficient is the leading coefficient. For \( 3x^2 + 2x - 5 \) the leading term is \( 3x^2 \) and the leading coefficient is \( 3 \). We usually write polynomials with the highest power first, so the leading term is right there at the start where it is easy to spot.

Polynomials also get friendly names based on how many terms they have:

• a monomial has one term, like \( 7x^3 \);
• a binomial has two terms, like \( x + 3 \);
• a trinomial has three terms, like \( 3x^2 + 2x - 5 \).

Beyond three terms we usually just say "polynomial" and let the count speak for itself.

Adding and subtracting: combine like terms

Here is the one idea behind all polynomial addition: you can only add terms that carry the same power of \( x \). We call these like terms. It is exactly the same instinct as money — you add dollars to dollars and cents to cents, never dollars to cents. Here, \( x^2 \)-terms add to \( x^2 \)-terms, \( x \)-terms to \( x \)-terms, constants to constants.

To add two polynomials, line them up by power and add each matching pair of coefficients. To subtract, do the same — but first flip the sign of every term you are subtracting.

Multiplying: the distributive law, then FOIL

Multiplication leans on one rule you have used since long division: the distributive law, which says \( a(b + c) = ab + ac \). To multiply a single term across a polynomial, you simply hand that term to every term inside the bracket:

Each piece picked up an extra \( x \) (so the powers all rose by one) and got multiplied by \( 2 \). That is the whole move.

Now, what if both factors are binomials, like \( (x + 3)(x - 2) \)? The same distributive law still applies — every term in the first bracket must meet every term in the second. With two terms each, that makes four products, and the handy reminder for them is FOIL: First, Outer, Inner, Last.

The quadratic and its parabola

The degree-2 polynomial earns a name of its own: the quadratic, written in general form as

Quadratics matter so much because of the picture they draw. Plot \( y = ax^2 + bx + c \) and you always get the same elegant U-shaped curve called a parabola. Three features tell you everything about it:

• The vertex is the turning point — the lowest point of the U (or the highest, if the U is flipped). It is where the curve stops falling and starts rising.
• The roots are the \( x \)-intercepts, where the curve crosses the \( x \)-axis. They are the values of \( x \) that make \( y = 0 \). A parabola can have two roots, one, or none.
• The sign of \( a \) sets the direction: if \( a > 0 \) the parabola opens upward (a valley), and if \( a < 0 \) it opens downward (a hill).

Drag the sliders below to watch \( y = ax^2 + bx + c \) change shape — this one is \( y = x^2 - 2x - 3 \), whose vertex sits at \( (1, -4) \) and whose roots are \( x = -1 \) and \( x = 3 \).

Look at how the picture matches the algebra. Because \( a = 1 > 0 \), the parabola opens upward. It crosses the \( x \)-axis twice — at \( x = -1 \) and \( x = 3 \) — and sure enough \( x^2 - 2x - 3 \) factors as \( (x + 1)(x - 3) \), which is zero at exactly those two values. Between the roots the curve dips down to its vertex at \( (1, -4) \), the bottom of the valley. Every quadratic you will study lives somewhere on this family of curves; changing \( a \), \( b \), and \( c \) just slides, stretches, and flips the same fundamental shape.

Practice

Try each one yourself, then reveal the full solution.