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Polynomials
Introduction
Geometrical Meaning of the Zeroes of a Polynomial
Practice: Zeroes from Graphs
Relationship between Zeroes and Coefficients of a Polynomial
Practice: Zeroes and Coefficients
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Glossary

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PolynomialsIntroduction

Reading time: ~25 min

Aman lofts a cricket ball high over the pitch. His cousin Riya, who loves physics, has a rule for it: t seconds after the shot, the ball is 20t - 5t^2 metres above the ground.

That one short rule describes the whole flight — how high the ball climbs, and when it comes back down.

Move the clock: t = ${t} seconds.

At what moment is the ball back on the ground?

At t = 4 the rule gives 20 \times 4 - 5 \times 4^2 = 80 - 80 = 0 (and it gave 0 at t = 0 too, when the ball left the bat). A rule like 20t - 5t^2, built from powers of a variable, is called a polynomial, and the inputs that make it equal to zero are called its zeroes. In this chapter you will hunt for zeroes, see them on graphs, and crack a secret code that predicts them.

Warm-up: What Counts as a Polynomial?

You met polynomials in Class IX. A polynomial in a variable x is built from x and ordinary numbers using only adding, subtracting and multiplying. That forces every power of x to be a whole number: 0, 1, 2, 3, …

The highest power that appears is called the degree of the polynomial. So 4x + 2 has degree 1, and 2y^2 - 3y + 4 (a polynomial in the variable y) has degree 2. Your turn:

  • 5x^3 - 4x^2 + x - \sqrt{2} has degree .
  • 7u^6 - \frac{3}{2}u^4 + 4u^2 + u - 8 has degree .

Some expressions only look like polynomials: \frac{1}{x-1}, \sqrt{x} + 2 and \frac{1}{x^2 + 2x + 3} are not polynomials.

The quick test

Write them with powers and you can see the problem: \frac{1}{x-1} = (x-1)^{-1} needs a negative power, and \sqrt{x} = x^{1/2} needs a fractional one. So if the variable ever sits under a root sign or in a denominator, the expression is not a polynomial. (Roots of plain numbers are fine — the \sqrt{2} above is just a number.)

Drag each expression into the right box.

4x + 2
√x + 2
5x³ − 4x² + x
1/(x − 1)
7u⁶ + 4u² − 8
1/(x² + 2x + 3)
A polynomial
Not a polynomial

Three Families

Polynomials are grouped into families by their degree. The three you will meet most often are:

  • Degree 1 — linear, such as 2x - 3, \sqrt{3}x + 5, y + \sqrt{2} or \frac{2}{3}u + 1. General form: ax + b.
  • Degree 2 — quadratic, such as 2x^2 + 3x - \frac{2}{5}, y^2 - 2 or 2 - x^2 + \sqrt{3}x. The name comes from the Latin quadratus, "square", because the top term is a square. General form: ax^2 + bx + c.
  • Degree 3 — cubic, such as 2 - x^3, \sqrt{2}x^3 or 3x^3 - 2x^2 + x - 1. General form: ax^3 + bx^2 + cx + d.

In these general forms, the letters a, b, c, d stand for numbers called coefficients. One rule matters a lot: the leading coefficient a must never be . The builder below shows why.

Change the coefficients with the + and − buttons, and complete all three missions.

Speed Round: Name the Family

Linear, quadratic, cubic — or not a polynomial at all? Look for the highest power, and watch out for variables hiding in denominators and under roots.

Get five right to move on.

The Polynomial Machine

Think of a polynomial as a machine: drop a number in, and a number comes out. Take p(x) = x^2 - 3x - 4. Drop in 2 and the machine works out p(2) = 2^2 - 3 \times 2 - 4 = . Drop in 0 and it returns p(0) = .

In general, if p(x) is a polynomial in x and k is any real number, the output you get by putting k in place of x is called the value of p(x) at x = k. We write it as p(k).

Feed the machine. Two of these inputs make it print 0 — find both.

Let's check those two special inputs by hand:

p(-1) = (-1)^2 - 3 \times (-1) - 4 =

p(4) = 4^2 - 3 \times 4 - 4 =

Since p(-1) = 0 and p(4) = 0, the numbers −1 and 4 are called the zeroes of the polynomial x^2 - 3x - 4. In general, a real number k is a zero of a polynomial p(x) if p(k) = 0.

Zero = the input that makes it vanish

Remember Aman's shot? The zeroes of 20t - 5t^2 are t = 0 and t = 4 — the two moments when the ball is at ground level. Zeroes are also called roots, and finding them is one of the most useful things you can do with a polynomial.

One-Line Zeroes

For a linear polynomial, finding the zero takes a single line — you did this in Class IX. If k is a zero of p(x) = 2x + 3, then 2k + 3 = 0, so k = -\frac{3}{2}.

The same works for every linear polynomial ax + b: setting ak + b = 0 gives k = -\frac{b}{a}. So

\text{zero of } ax + b = -\frac{b}{a} = \frac{-(\text{constant term})}{\text{coefficient of } x}

Speed round: pick the zero of each linear polynomial. Get five right to move on.

Notice what just happened: you found every zero by reading nothing but the . Can the zeroes of a quadratic be read off its coefficients too? Keep that question in mind — we will crack it in the last section. First, let's learn to see zeroes.

Sina