Polynomials

Why This Matters

Before we start, a few words from earlier classes turn up again and again here — variable, coefficient, constant term. A 30-second refresher so they’re fresh:

A polynomial is an expression made from a variable (like x) using only three things: adding, subtracting, and whole-number powers. For example, x² − 3x − 4 is a polynomial. Polynomials show up everywhere in maths and science. The path of a ball you throw is a polynomial. So is the area of a square garden as it grows bigger. So is the profit of a shop as it changes its price. They are one of the most useful tools you will learn.

The biggest question we ask about any polynomial is simple: for what value of x does it become 0? The x-values that make this happen are called its zeroes. Zeroes are hiding inside many real problems. The moment a ball hits the ground is a zero. The price at which a business neither gains nor loses money is a zero. Solving a real problem very often means finding a zero.

In this chapter you’ll see two lovely ideas. First, zeroes have a picture: they are the exact spots where the polynomial’s graph crosses the x-axis. Second, the zeroes are secretly connected to the coefficients (the numbers in front of x). Without solving anything, those numbers already tell you the sum and product of the zeroes. This hidden link will help you again and again.

The Big Idea

A zero of a polynomial p(x) is a value of x that makes p(x) equal to 0. On a graph, the zeroes are the x-intercepts — the points where the curve y = p(x) touches or crosses the x-axis. The zeroes are also linked to the coefficients. For a quadratic ax² + bx + c, the sum of the zeroes is −b/a and the product of the zeroes is c/a.

Let’s Break It Down

Polynomials, degree and zeroes

The degree of a polynomial is the highest power of the variable in it. For example, in x² − 3x − 4 the highest power is 2, so its degree is 2. We give polynomials names based on their degree:

(In every case a ≠ 0. If a were 0, the highest term would disappear and the degree would drop.) Expressions like 1/x or √x + 2 are not polynomials. This is because the powers in a polynomial must be whole numbers. Here 1/x means x to the power −1, and √x means x to the power 1/2 — neither is a whole number.

The value of p(x) at x = k is written p(k). You get it by putting k in place of x everywhere in the polynomial. We say a number k is a zero of p(x) when p(k) = 0. For example, take p(x) = x² − 3x − 4:

• p(−1) = (−1)² − 3(−1) − 4 = 1 + 3 − 4 = 0, and
• p(4) = 16 − 12 − 4 = 0,

Both inputs give an output of 0, so −1 and 4 are the zeroes of x² − 3x − 4.

For a linear polynomial ax + b, we find the zero by setting ax + b = 0 and solving for x. This gives one zero: x = −b/a. Notice that the zero is built from the coefficients a and b. That is an early hint that zeroes and coefficients are linked.

The geometrical meaning of a zero

To picture zeroes on a graph, you first need to be comfortable reading points off the x and y axes. Here’s a quick reminder of how that works:

Here is the picture that makes everything clear. Suppose you draw the graph of y = p(x). The zeroes are the x-values where the graph crosses or touches the x-axis. Why? On the x-axis, y is always 0. So at those points y = p(x) = 0, which is exactly what a zero means.

A quadratic always makes a curve called a parabola — a smooth U-shape. It opens upward when a > 0 and downward when a < 0. Figure 2.1 below shows how the zeroes of x² − 3x − 4 (which we found to be −1 and 4) show up as the two points where the curve meets the x-axis:

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A parabola can sit against the x-axis in only three ways. So a quadratic can have either two, one, or no real zeroes. Figure 2.2 below lays out all three side by side:

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This picture explains a key fact:

A polynomial of degree n has at most n zeroes. So a quadratic (degree 2) has at most 2 zeroes, and a cubic (degree 3) has at most 3.

The graph shows this is believable — a parabola can only meet the axis up to twice. But why is the limit exactly the degree, and never more? Here is the real reason, and it doesn’t need a graph at all.

Each different zero quietly “uses up” one factor. We’ll prove this properly in a moment (it’s called the factor theorem), but the idea is short: if α is a zero, then (x − α) is a factor of the polynomial. Two different zeroes mean two different factors, like (x − α)(x − β). And factors multiply together, so their degrees add up — two linear factors make degree 2, three make degree 3, and so on.

Now watch what goes wrong if there were too many zeroes. Suppose a quadratic — degree 2 — had 3 different zeroes. Then it would have 3 factors, (x − α)(x − β)(x − γ), which is degree 3. But the polynomial is only degree 2. A polynomial can’t be degree 2 and degree 3 at the same time. So this is impossible. The most zeroes a degree-2 polynomial can have is 2. The same reasoning caps a cubic at 3, and a degree-n polynomial at n. Figure 2.3 below walks through this argument:

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Let’s put that “at most n” rule to work on a cubic whose graph cuts the axis three times:

Zeroes are tied to the coefficients

Now for the surprise. Take any quadratic and factorise it. For example, p(x) = 2x² − 8x + 6 = 2(x − 1)(x − 3), so its zeroes are 1 and 3. Now look at what the coefficients already tell us:

• Sum of zeroes = 1 + 3 = 4. And −b/a = −(−8)/2 = 4. Same answer.
• Product of zeroes = 1 × 3 = 3. And c/a = 6/2 = 3. Same answer.

This is not luck. It is true for every quadratic, and we can prove it.

The proof leans on one quiet fact we keep using: if α is a zero of a polynomial, then (x − α) is a factor of it. This is called the factor theorem. It sounds like a big claim, so let’s see exactly why it must be true — it comes straight from ordinary division.

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With the factor theorem in hand, the proof of the sum and product rules falls out — its very first step is just to write the zeroes as factors:

That whole proof is just one simple idea. We wrote the same quadratic in two ways, placed them one above the other, and matched the matching terms. This is much easier to see in a picture than to read in words. Figure 2.5 below lines the two forms up column by column:

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An easy way to remember it: sum = −(coefficient of x)/(coefficient of x²) and product = (constant)/(coefficient of x²).

There is also a nice way to see both rules right on the graph. A parabola is symmetric — its left and right halves are mirror images. So its two zeroes sit at equal distances from the centre line, called the axis of symmetry. That centre line is at x = −b/2a, which is the average (the middle) of the two zeroes. If you double the average, you get the sum, which is −b/a. Also, the curve crosses the y-axis at the value c. Since c = a·αβ, dividing by a gives the product c/a. Figure 2.6 below shows both rules on the actual graph:

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Let’s test these two rules on a fresh quadratic — find its zeroes, then check the sum and product against the coefficients:

This link works both ways. If someone gives you the sum and product of the zeroes, you can build the quadratic back. A polynomial whose zeroes add up to S and multiply to P is x² − Sx + P (here we take a = 1).

Let’s use that to build a quadratic when we’re only told the sum and product of its zeroes:

The same idea for cubics

A cubic ax³ + bx² + cx + d with zeroes α, β, γ follows three matching rules. We prove them the same way as before — by expanding a(x − α)(x − β)(x − γ) and matching terms:

• α + β + γ = −b/a (sum of zeroes)
• αβ + βγ + γα = c/a (sum of products, two at a time)
• αβγ = −d/a (product of zeroes)

Notice the pattern in the signs: −b/a, then +c/a, then −d/a. The sign keeps flipping (minus, plus, minus) as you move down the coefficients.

Figure 2.7 below shows where all three cubic rules come from — it’s the very same matching trick, just one degree higher:

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Common Mistakes

Quick Check

Practice Problems

Easy

Medium

These next ones don’t factorise at a glance, so we’ll lean on “splitting the middle term” from Class 9. Here’s how that method works:

The next problem has no middle term to split, but a square-minus-a-square identity unlocks it. A quick reminder of that identity:

Challenge

Summary

You should now be able to explain:

• A polynomial’s degree is its highest power. Degrees 1, 2, 3 are called linear, quadratic, cubic. A quadratic has the form ax² + bx + c with a ≠ 0.
• A zero is an input k that makes p(k) = 0. It is not the same as p(0).
• On a graph, the zeroes of p(x) are the x-values where y = p(x) meets the x-axis.
• A quadratic’s graph is a parabola (opens up if a > 0, down if a < 0). It has two, one or no real zeroes. In general, a polynomial of degree n has at most n zeroes.
• For a quadratic ax² + bx + c: sum of zeroes = −b/a and product = c/a. (You can prove both by expanding a(x − α)(x − β).)
• The other way round: a quadratic with sum S and product P is x² − Sx + P.
• For a cubic ax³ + bx² + cx + d: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.

What’s Next

You now know that zeroes are where one graph meets the x-axis. Next, in Pair of Linear Equations in Two Variables, we draw two straight-line graphs on the same axes. Then we ask where they meet each other. That meeting point is the one solution that fits both equations at the same time. It is the same idea — picture meeting algebra — taken one step further.