01 · REMAINDER THEOREM
The Remainder Theorem
Let p(x) be any polynomial of degree greater than or equal to 1 and let a be any real number. If p(x) is divided by the linear polynomial (x - a), then the remainder is p(a).
SOLVED EXAMPLE
Find the remainder when p(x) = x³ - 3x² + 4x + 50 is divided by (x - 3)
By Remainder Theorem, zero of (x - 3) is x = 3. Substitute x = 3 into p(x):
p(3) = (3)³ - 3(3)² + 4(3) + 50
= 27 - 27 + 12 + 50 = 62
= 27 - 27 + 12 + 50 = 62
The remainder is 62.
02 · FACTOR THEOREM
The Factor Theorem
If p(x) is a polynomial of degree n ≥ 1 and a is any real number, then:
1. (x - a) is a factor of p(x) if p(a) = 0.
2. p(a) = 0 if (x - a) is a factor of p(x).
FACTOR THEOREM EXAMPLE
Find the value of k if (x - 1) is a factor of p(x) = 4x³ + 3x² - 4x + k
Since (x - 1) is a factor, p(1) must be equal to 0:
p(1) = 4(1)³ + 3(1)² - 4(1) + k = 0
4 + 3 - 4 + k = 0
3 + k = 0 ⟹ k = -3
4 + 3 - 4 + k = 0
3 + k = 0 ⟹ k = -3