(i) x3 + x2 + x + 1
⇒ x + 1 = 0
⇒ x = -1
p(x) = x3 + x2 + x + 1
p(-1) = (-1)3 + (-1)2 + (-1) + 1
= -1 + 1 -1 + 1
= 0
Remainder is zero (0), so (x + 1) is factor of this polynomial.
(ii) x4 + x3 + x2 + x + 1
⇒ x + 1 = 0
⇒ x = -1
p(x) = x4 + x3 + x2 + x + 1
p(-1) = (-1)4 + (-1)3 + (-1)2 + (-1) + 1
= 1 -1 + 1 -1 + 1
= 1
Remainder is not zero (0), so (x + 1) is not a factor of this polynomial.
(iii) x4 + 3x3 + 3x2 + x + 1
⇒ x + 1 = 0
⇒ x = -1
p(x) = x4 + 3x3 + 3x2 + x + 1
p(-1) = (-1)4 + 3 x (-1)3 + 3 x (-12) + (-1) + 1
= 1 -3 + 3 -1 + 1
= 1
Remainder is not zero (0), so (x + 1) is not a factor of this polynomial.
(iv) x3 - x2 - (2 + )x +
⇒ x + 1 = 0
⇒ x = -1
p(x) = x3 - x2 - (2 + )x +
p(-1) = (-1)3 - (-1)2 - (2 + )(-1) +
= -1 - 1 + 2 + +
= -2 + 2 + 2
= 2
Remainder is not zero (0), so (x + 1) is not a factor of this polynomial.
Polynomials — Interactive Study Guide
Master polynomial basics, Remainder and Factor Theorems, factorisation, and algebraic identities.
Polynomial Classification
Not a polynomial: √x (fractional power), 1/x = x−1 (negative power), x + 1/x.
Polynomial: 5 (constant), 3x + 2 (linear), x² − 1 (quadratic), 2x³ + x − 1 (cubic).
Remainder and Factor Theorems — Quick Guide
Factor Theorem: (x − a) is a factor of p(x) ⇔ p(a) = 0.
Watch the sign! Dividing by (x + 3) means a = −3. So remainder = p(−3).
Identity Mastery Checklist
| See This Pattern | Use This Identity |
|---|---|
| a² + 2ab + b² | = (a + b)² |
| a² − 2ab + b² | = (a − b)² |
| a² − b² | = (a + b)(a − b) |
| a³ + b³ | = (a + b)(a² − ab + b²) |
| a³ − b³ | = (a − b)(a² + ab + b²) |
| a + b + c = 0 | ⇒ a³ + b³ + c³ = 3abc |
Quick Self-Check
- Degree of 5x³ − 2x + 1? (3)
- Remainder when x² + 3x + 2 is divided by (x + 1)? (p(−1) = 1 − 3 + 2 = 0)
- Expand: (2a + 3b)² (= 4a² + 12ab + 9b²)
- Factorise: 8x³ − 27 (= (2x − 3)(4x² + 6x + 9))