Chapter 2 ยท Introduction

Chapter 2 โ€” Polynomials

Polynomials describe the flight of a cricket ball, the curve of an arch and the profit of a stall. Their zeroes answer the questions that matter โ€” where does it land, how wide is it, when do we break even?

A batter lofts the ball towards the boundary. For a second or two everyone in the stadium is asking the same question: will it carry for six? The height of the ball above the ground, measured against the distance it has travelled, follows a curve that is very close to a quadratic polynomial $h(x)=ax^2+bx+c$. The ball lands where its height becomes $0$ โ€” at a zero of the polynomial. If that zero is beyond the boundary rope, it's a six.

In this chapter you learn to read polynomials like this. First we sort polynomials by their degree โ€” linear, quadratic, cubic โ€” and find their values and zeroes. Then we see the picture behind the algebra: the zeroes of $p(x)$ are exactly the points where the graph of $y=p(x)$ meets the $x$-axis, so a quadratic has at most two zeroes and a cubic at most three. Finally we discover that the zeroes of a quadratic are hidden in its coefficients: their sum is $-\tfrac ba$ and their product is $\tfrac ca$. That lets us check zeroes in seconds and build a quadratic with any zeroes we choose.

Polynomials and their degreeโ†’Values and zeroes: $p(k)=0$โ†’Zeroes are where the graph meets the $x$-axisโ†’$\alpha+\beta=-\tfrac ba$, $\;\alpha\beta=\tfrac ca$โ†’Build a quadratic from its zeroes
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • decide whether an expression is a polynomial, and find its degree and type (linear, quadratic, cubic);
  • find the value $p(k)$ of a polynomial and check whether $k$ is a zero;
  • read the number of zeroes and their values from the graph of $y=p(x)$, and explain why a polynomial of degree $n$ has at most $n$ zeroes;
  • find the zeroes of a quadratic polynomial by splitting the middle term and verify that $\alpha+\beta=-\tfrac ba$ and $\alpha\beta=\tfrac ca$;
  • build a quadratic polynomial from the sum and product of its zeroes, and evaluate expressions such as $\alpha^2+\beta^2$ without finding the zeroes.
Teaser โ€” Six or Not? Find the Zero

Each shot follows a quadratic polynomial $h(x)$: the height in metres after the ball has travelled $x$ metres. The ball lands at the zero of $h(x)$. The boundary rope is $65$ m away. Press Play to watch the shots one after another, or pick one and predict first: is its zero beyond $65$?

height polynomialโ€”
ball atโ€”
zero (landing point)โ€”

Sections in this chapter

2.1

Polynomials and the Geometrical Meaning of Zeroes

Degree, value and zeroes of a polynomial; zeroes as the points where a line, parabola or cubic meets the $x$-axis.

$p(k)=0\iff(k,0)$ on the graph
Open section โ†’2.2

Relationship between Zeroes and Coefficients

Sum and product of the zeroes of a quadratic, building a quadratic from its zeroes, and expressions like $\alpha^2+\beta^2$.

$\alpha+\beta=-\tfrac ba,\ \ \alpha\beta=\tfrac ca$
Open section โ†’
โœ… Before you start
From Class 9 Polynomials you need: the words term, coefficient and degree; finding a value such as $p(2)$; the Factor Theorem (if $p(a)=0$ then $x-a$ is a factor of $p(x)$); factorising quadratics such as $x^2+5x+6$ and $2x^2-7x+3$ by splitting the middle term; and the identities $(a+b)^2=a^2+2ab+b^2$ and $a^2-b^2=(a+b)(a-b)$. From Class 9 Linear Equations in Two Variables you need to plot points and draw the graph of a line such as $y=2x+3$. Warm up: factorise $x^2-x-6$ and say for which $x$ it equals $0$ (answer: $(x-3)(x+2)$, so $x=3$ or $x=-2$).
โ† 1.3 Revisiting Irrational Numbers2.1 Polynomials and the Geometrical Meaning of Zeroes โ†’