Exercise 2, Questions 3–4 – Solutions p(1), p(10), p(0), p(−1)
Nine polynomials to evaluate. Work each one in your notebook first, then step through the animation: substitute in brackets, do the powers, multiply, add. Look out for the patterns from page 10.
- p(0) = constant term; p(1) = sum of the coefficients; p(−1): odd-power coefficients change sign; p(10) = the coefficients as digits.
- (−1)² = 1, (−1)³ = −1.
Question 3 · p(1) and p(10)
Q3. ചുവടെയുള്ള ബഹുപദങ്ങളിൽ, p(1) ഉം p(10) ഉം കണക്കാക്കുക. For each polynomial, find p(1) and p(10).
(i) p(x) = 2x + 5 (ii) p(x) = 3x² + 6x + 1 (iii) p(x) = 4x³ + 2x² + 3x + 7
10² = 100 and 10³ = 1000. Compare each p(10) with the coefficients – do you see them?
(i) p(x) = 2x + 5
(ii) p(x) = 3x² + 6x + 1
(iii) p(x) = 4x³ + 2x² + 3x + 7
| p(x) | p(1) | p(10) | |
|---|---|---|---|
| (i) | 2x + 5 | 7 | 25 |
| (ii) | 3x² + 6x + 1 | 10 | 361 |
| (iii) | 4x³ + 2x² + 3x + 7 | 16 | 4237 |
Every number we write is a polynomial evaluated at 10 – that is what place value means. Seeing this makes polynomial arithmetic feel like ordinary arithmetic with “carrying” switched off.
Question 4 · p(0), p(1) and p(−1)
Q4. ചുവടെയുള്ള ബഹുപദങ്ങളിൽ, p(0), p(1), p(−1) ഇവ കണക്കാക്കുക. For each polynomial, find p(0), p(1) and p(−1).
(i) 3x + 5 (ii) 5x − 8 (iii) 3x² + 6x + 1 (iv) 2x² − 5x + 3 (v) 4x³ + 2x² + 3x + 7 (vi) ax³ + bx² + cx + d
Put negative numbers in brackets: 5(−1) − 8, 2(−1)² − 5(−1) + 3. For (vi), do exactly the same with letters.
(i) p(x) = 3x + 5
(ii) p(x) = 5x − 8
(iii) p(x) = 3x² + 6x + 1
(iv) p(x) = 2x² − 5x + 3
(v) p(x) = 4x³ + 2x² + 3x + 7
(vi) p(x) = ax³ + bx² + cx + d
| p(x) | p(0) | p(1) | p(−1) | |
|---|---|---|---|---|
| (i) | 3x + 5 | 5 | 8 | 2 |
| (ii) | 5x − 8 | −8 | −3 | −13 |
| (iii) | 3x² + 6x + 1 | 1 | 10 | −2 |
| (iv) | 2x² − 5x + 3 | 3 | 0 | 10 |
| (v) | 4x³ + 2x² + 3x + 7 | 7 | 16 | 2 |
| (vi) | ax³ + bx² + cx + d | d | a + b + c + d | −a + b − c + d |
p(0), p(1), p(−1) are quick checks: after expanding a product like (x + 1)(x + 2)(x + 3), put x = 1 into both forms – if the answers differ, there is a mistake. And a number that makes p(x) zero (like 1 in (iv)) will be important in later classes.