1. Terms
- Degree = highest power. Linear (1), quadratic (2), cubic (3).
- A zero of p(x) is a value with p(k) = 0. A degree-n polynomial has at most n zeros.
2. Remainder and factor theorems
- Remainder theorem: p(x) ÷ (x − a) leaves remainder p(a). For (ax − b), remainder = p(b/a).
- Factor theorem: (x − a) is a factor ⇔ p(a) = 0.
(ASO 2015) x⁴ + x³ − 2x² + x + 1 ÷ (x − 1): p(1) = 2.
(ASO 2015) (x − 1) a factor of 4x³ + 3x² − 4x + k → 3 + k = 0 → k = −3.
3. Zeros and coefficients
| Polynomial | Relations |
|---|---|
| ax² + bx + c | α + β = −b/a, αβ = c/a |
| ax³ + bx² + cx + d | α + β + γ = −b/a; αβ + βγ + γα = c/a; αβγ = −d/a |
(ASO 2015) Sum of squares of zeros of x² − 8x + k is 40 → 64 − 2k = 40 → k = 12.
4. Identities (most used)
- (a ± b)² = a² ± 2ab + b²; a² − b² = (a + b)(a − b)
- (a + b)² − (a − b)² = 4ab; (a + b)² + (a − b)² = 2(a² + b²)
- (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
- (a ± b)³ = a³ ± b³ ± 3ab(a ± b)
- a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²)
- a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca). If a + b + c = 0, a³ + b³ + c³ = 3abc.
5. Solved examples
a + b = 7, ab = 10 → a² + b² = 49 − 20 = 29; a³ + b³ = 343 − 210 = 133.
x + 1/x = 4 → x² + 1/x² = 14; x³ + 1/x³ = 64 − 12 = 52.
(−12)³ + 7³ + 5³ = 3 × (−12) × 7 × 5 = −1260 (since −12 + 7 + 5 = 0).
Dividing a degree-5 polynomial by a degree-2 polynomial gives a quotient of degree 3 (n − m).
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
Basic level
Example 1. The remainder when x^4 + x^3 − 2x^2 + x + 1 is divided by (x + 1) is:
(a) 0 (b) -2 (c) -4 (d) -1
Answer: (b) -2
Solution: Remainder theorem: put x = -1. Value = -2.
Example 2. If p(x) = 2x³ − 5x − 1, then p(-1) = ?
(a) 2 (b) 6 (c) -2 (d) 4
Answer: (a) 2
Solution: p(-1) = 2(-1)³ + -5(-1) + -1 = 2.
Example 3. If p(x) = 3x³ + 6x − 6, then p(-1) = ?
(a) 15 (b) -9 (c) -15 (d) -13
Answer: (c) -15
Solution: p(-1) = 3(-1)³ + 6(-1) + -6 = -15.
ASO exam level
Example 4. The sum of the squares of the zeroes of x² − 6x + k is 52. The value of k is:
(a) -8 (b) 8 (c) -6 (d) -16
Answer: (a) -8
Solution: α² + β² = (α+β)² − 2αβ = 36 − 2k = 52 → k = -8.
Example 5. The sum of the squares of the zeroes of x² − 6x + k is 52. The value of k is:
(a) -6 (b) -16 (c) -8 (d) 8
Answer: (c) -8
Solution: α² + β² = (α+β)² − 2αβ = 36 − 2k = 52 → k = -8.
Example 6. If (x − 1) is a factor of 1x³ + 2x² − 5x + k, then k = ?
(a) 2 (b) -2 (c) 3 (d) 0
Answer: (a) 2
Solution: Put x = 1: 1 + 2 + -5 + k = 0 → k = 2.
Example 7. The remainder when x³ − 1x² + 4x − 5 is divided by (x − 4) is:
(a) 60 (b) 59 (c) -5 (d) -101
Answer: (b) 59
Solution: Remainder theorem: put x = 4: 59.
Example 8. The remainder when x³ − 1x² − 4x + 4 is divided by (x − 1) is:
(a) 1 (b) 6 (c) 0 (d) 4
Answer: (c) 0
Solution: Remainder theorem: put x = 1: 0.
Example 9. If a + b = 12 and ab = 20, then a³ + b³ = ?
(a) 1,488 (b) 2,448 (c) 1,008 (d) 1,728
Answer: (c) 1,008
Solution: a³ + b³ = (a + b)³ − 3ab(a + b) = 1728 − 720 = 1008.
Advanced level
Example 10. If α and β are zeroes of x² − 7x + 10, then 1/α + 1/β = ?
(a) 70 (b) 10/7 (c) 17 (d) 7/10
Answer: (d) 7/10
Solution: 1/α + 1/β = (α + β)/αβ = 7/10.
Example 11. If α and β are zeroes of x² − 3x + 2, then 1/α + 1/β = ?
(a) 2/3 (b) 6 (c) 3/2 (d) 5
Answer: (c) 3/2
Solution: 1/α + 1/β = (α + β)/αβ = 3/2.
Example 12. If (x − 4) is a factor of x³ + 4x² + 2x + k, then k = ?
(a) -138 (b) -136 (c) 136 (d) -135
Answer: (b) -136
Solution: p(4) = 0 → 64 + 64 + 8 + k = 0 → k = -136.
Example 13. If (x + 3) is a factor of x³ − 1x² − 2x + k, then k = ?
(a) -30 (b) 31 (c) 30 (d) 28
Answer: (c) 30
Solution: p(-3) = 0 → -27 + -9 + 6 + k = 0 → k = 30.
Topic-wise practice for Polynomials and identities (Quant): Algebraic identities, remainder and factor theorem. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Algebraic identities: (a + b)², (a − b)², a² − b², a³ ± b³
- Find x³ + 1/x³ from x + 1/x
- Remainder theorem
- Factor theorem: find k
- Zeros of a polynomial; sum and product
- Degree and value of a polynomial