Polynomials & Degree സവിശേഷ വാചകങ്ങൾ
The perimeter, area and volume rules of the last pages – 4x + 10, x² + 5x + 6, x³ + 6x² + 11x + 6 – all have the same kind of build. Expressions built that way get a special name: polynomials (ബഹുപദങ്ങൾ). On this page you will learn to recognise them, to spot expressions that are not polynomials, and to read off a polynomial's degree and coefficients.
- Powers: x² = x × x, x³ = x × x × x.
- p(x) = 4x + 10, a(x) = x² + 5x + 6, v(x) = x³ + 6x² + 11x + 6 from the growing rectangle and the growing block.
- Pythagoras: the hypotenuse of a right triangle with legs a, b is √(a² + b²); an isosceles right triangle with legs a has hypotenuse a√2.
1. Rules are operations on numbers
We wrote p(x) = 4x + 10 for the perimeter of a growing rectangle. But forget the rectangle for a moment: the rule simply says “multiply the number by 4 and add 10”. It is an operation on plain numbers – like a machine that takes in a number and gives out another.
In all these rules, powers of the changing number x are multiplied by fixed numbers, and these products are added; a fixed number may also be added. An algebraic expression (ബീജഗണിത വാചകം) that involves only such operations is called a polynomial (ബഹുപദം).
“Poly” means many and “nomial” means terms: a polynomial is a sum of many terms, each term being (fixed number) × (a power of x). The Malayalam ബഹുപദം says exactly the same: ബഹു = many, പദം = term.
Think of x, x², x³, … as LEGO bricks of different sizes. A polynomial is any model you can build by taking some number of each brick (the fixed numbers) and putting them together (adding). You may not saw a brick into a square root or turn it upside down into 1/x – those are not allowed moves.
2. A measure that is not a polynomial – the diagonal
Take rectangles whose one side is 1 cm longer than the other. If the shorter side is x cm, the longer is x + 1 cm and, by Pythagoras, the diagonal (വികർണം) is
√(x² + (x + 1)²) = √(2x² + 2x + 1) cm
Inside the root, 2x² + 2x + 1 is a polynomial. But then we take the square root of a changing number – that is not one of the allowed operations.
√(2x² + 2x + 1) takes the square root (വർഗമൂലം) of an expression that changes with x. So by our definition, the diagonal is not a polynomial in x.
3. √2 is allowed – as a fixed number
Here is the figure of the textbook: an isosceles right triangle (സമപാർശ്വമട്ടത്രികോണം) with legs 4 cm, and a rectangle 1 cm wide attached along its hypotenuse (കർണം). What is the area of the whole shape?
The area ½x² + √2 x contains √2, but √2 is a fixed number – it never changes when x changes. The only things done to x are squaring it and multiplying by the fixed numbers ½ and √2. So ½x² + √2 x is a polynomial. What matters is what happens to x, not what the fixed numbers look like.
4. Another non-polynomial – dividing by x
Rectangles of area 25 sq cm: if one side is x cm, the other is 25/x cm and the perimeter is
2x + 2 × 25/x = 2x + 50/x
50/x = 50 × (1/x) uses the reciprocal (വ്യുൽക്രമം) of the changing number. So 2x + 50/x is not a polynomial.
5. Polynomial or not? – the sorting game
Decide for each expression. The test: is anything done to x other than taking powers (x, x², x³, …), multiplying by fixed numbers and adding?
6. Degree – കൃത്യങ്കം
In a polynomial we take powers of the changing number. The largest exponent that appears is called the degree of the polynomial (ബഹുപദത്തിന്റെ കൃത്യങ്കം).
- p(x) = 4x + 10 → degree 1 → a first degree polynomial (ഒന്നാംകൃതി ബഹുപദം)
- a(x) = x² + 5x + 6 → degree 2 → a second degree polynomial (രണ്ടാംകൃതി ബഹുപദം)
- v(x) = x³ + 6x² + 11x + 6 → degree 3 → a third degree polynomial (മൂന്നാംകൃതി ബഹുപദം)
Remember 4x means 4x¹ – its exponent is 1; and the lone 10 has no x at all (think of it as 10x⁰).
7. General forms and coefficients – ഗുണകങ്ങൾ
| Degree | Name | General form | Example |
|---|---|---|---|
| 1 | first degree ഒന്നാംകൃതി | ax + b | 4x + 10 (a = 4, b = 10) |
| 2 | second degree രണ്ടാംകൃതി | ax² + bx + c | x² + 5x + 6 (a = 1, b = 5, c = 6) |
| 3 | third degree മൂന്നാംകൃതി | ax³ + bx² + cx + d | x³ + 6x² + 11x + 6 |
Here a, b, c, d stand for fixed numbers: they do not change when we take different numbers as x. They are called the coefficients (ഗുണകങ്ങൾ) of the respective powers. They can be any real numbers – whole numbers, fractions, negatives, even √2.
- In ax² + bx + c we need a ≠ 0 – otherwise there is no x² term and the degree drops to 1.
- The term without x (b in ax + b, c in ax² + bx + c, d in ax³ + …) is called the constant term.
- A number alone, like 7, is also a polynomial – of degree 0 (7 = 7x⁰).
Slide a to 0 and watch the degree fall from 2 to 1. Negative coefficients turn “+” into “−”.