Summary & Quick Quiz ബഹുപദങ്ങൾ
Every rule of the chapter on one page, the mistakes to avoid, flashcards, a scored quiz and your study checklist.
1. The chapter in one picture
A 3 cm × 2 cm rectangle with every side increased by x. Its perimeter is p(x) = 4x + 10 (degree 1) and its area is a(x) = x² + 5x + 6 (degree 2). The block 1 × 2 × 3 grown the same way has v(x) = x³ + 6x² + 11x + 6 (degree 3).
2. The key ideas
When every side of the 3 × 2 rectangle grows by x cm, the perimeter is p(x) cm. Writing p(x) instead of p shows that p changes with x; p(3) = 22 means “x = 3 gives p = 22”. (അളവുകളുടെ ബീജഗണിതം)
Multiply out the brackets (ഗുണനസമവാക്യങ്ങൾ), letters first: (x + 3)(x + 2) = x² + 5x + 6; a(1½) = 15¾.
ഇത്തരം ക്രിയകൾ മാത്രം ഉൾപ്പെടുന്ന ബീജഗണിത വാചകത്തിന്റെ പൊതുവായ പേരാണ് ബഹുപദം.
An algebraic expression in which powers of the changing number x are multiplied by fixed numbers and added (a fixed number may be added too) is a polynomial (ബഹുപദം).
Square root of x (വർഗമൂലം) or reciprocal of x (വ്യുൽക്രമം) → not a polynomial. But roots of fixed numbers are fine: ½x² + √2 x is a polynomial.
ഇങ്ങനെ വരുന്ന ഏറ്റവും വലിയ കൃത്യങ്കത്തെ ബഹുപദത്തിന്റെ കൃത്യങ്കം എന്നാണ് പറയുന്നത്.
The highest power of x is the degree (കൃത്യങ്കം): 4x + 10 → 1, x² + 5x + 6 → 2, x³ + 6x² + 11x + 6 → 3.
a, b, c, d are fixed numbers, the coefficients (ഗുണകങ്ങൾ) of the powers; they can be any real numbers (a ≠ 0 for the stated degree).
p(0) is the constant term, p(1) the sum of the coefficients, and p(10) writes single-digit coefficients as digits: 4x³ + 2x² + 3x + 7 → 4237.
3. Formula sheet
| Situation | Rule | Degree | Example value |
|---|---|---|---|
| 3 × 2 rectangle, sides + x: perimeter | p(x) = 4x + 10 | 1 | p(1½) = 16 |
| 3 × 2 rectangle, sides + x: area | a(x) = x² + 5x + 6 | 2 | a(2) = 20 |
| 1 × 2 × 3 block, edges + x: volume | v(x) = x³ + 6x² + 11x + 6 | 3 | v(1) = 24 |
| Sides x and x + 1: perimeter, area | 4x + 2, x² + x | 1, 2 | a(3) = 12 |
| Box from 7 × 5 card, cut x | v(x) = 4x³ − 24x² + 35x | 3 | v(1) = 15 |
| Rectangle from a 1 m rope, side x | a(x) = 50x − x² | 2 | a(10) = a(40) = 400 |
| Path 1 m wide around a square of side x | 4x + 4 | 1 | x = 10: 44 sq m |
| (x + a)(x + b) | x² + (a + b)x + ab | 2 | (x + 5)(x + 3) = x² + 8x + 15 |
- p(2) is not p × 2. p(2) means “the value of p when x is 2”: for p(x) = 4x + 10, p(2) = 18.
- Forgetting brackets with negatives. For p(x) = 3x² + 6x + 1, p(−1) = 3(−1)² + 6(−1) + 1 = 3 − 6 + 1 = −2; writing −1² as −1 gives a wrong answer.
- Thinking any √ makes it “not a polynomial”. √2 x is a polynomial – √2 is a fixed number. Only roots of the changing number spoil it.
- Calling x + 1/x a polynomial. 1/x is the reciprocal of x – not a power of x we allow. Not a polynomial.
- Losing the sign of a coefficient. In 2x² − 5x + 3 the coefficient of x is −5, not 5.
- Degree = number of terms? No: x³ + 1 has two terms but degree 3.
- (x + 3)(x + 2) ≠ x² + 6. Multiply every term by every term: x² + 2x + 3x + 6 = x² + 5x + 6.
4. Flashcards
Click a card to flip it.
5. Scored quiz
Answer all ten, then press Check my answers. Each wrong answer shows the page to revise.
6. Your study checklist
Next: the model test paper ▶ – attempt it on paper first, then check with the solved paper.