Chapter 2: Polynomials

A polynomial is one of the friendliest objects in algebra: just a sum of powers of $x$ with numerical coefficients. Yet hidden inside polynomials are deep ideas , about graphs, about roots, about symmetry , that fuel almost every later chapter, from quadratic equations to coordinate geometry.

This chapter has two big themes. (i) The zeroes (or roots) of a polynomial: the values of $x$ for which $p(x) = 0$. Geometrically, these are exactly the $x$-coordinates where the graph $y = p(x)$ crosses the $x$-axis. (ii) The relationship between zeroes and coefficients: without solving the polynomial, just by looking at its coefficients, we can read off the sum and product of its zeroes.

For the board exam, this chapter is short but highly testable. Expect $1$-mark MCQs on degree and zeroes, $2$-mark questions on Vieta-style sum/product, $3$-mark problems on constructing a quadratic with given zeroes or finding the zeroes from coefficients, and small graph-reading questions on counting zeroes from a sketch.

Real-world hook? Polynomials model trajectories, profit curves, growth, and shapes of suspension cables. Every parabola of a thrown ball, every curve on a roller-coaster diagram, is a polynomial in disguise. Learning to read them is a first step into physics, engineering, and computer graphics.

What's inside

• Polynomials: definitions and degree , what they are, how we classify them.
• Geometrical meaning of zeroes , reading zeroes from a graph.
• Zeroes and coefficients of a quadratic , sum $-b/a$ and product $c/a$.
• Zeroes and coefficients of a cubic , the three classical identities.
• Constructing a polynomial from given zeroes , the reverse problem.

Key results / Formula card

• A polynomial of degree $n$ has at most $n$ zeroes (real).
• A quadratic with zeroes $\alpha, \beta$ is (up to a constant) $x^2 - (\alpha + \beta) x + \alpha \beta$.
• A cubic with zeroes $\alpha, \beta, \gamma$ is (up to a constant) $x^3 - (\sum \alpha) x^2 + (\sum \alpha\beta) x - \alpha\beta\gamma$.