FORMULA CHAPTER
Dividend = Divisor × Quotient + Remainder
Statement: When a polynomial f(x) is divided by x − a, the remainder is the value f(a).
Remainder = f(a) when f(x) is divided by (x − a)
Statement: The expression x − a is a factor of polynomial f(x) if and only if substituting x = a gives zero.
(x − a) is a factor of f(x) ⇔ f(a) = 0
Dividend = Divisor × Quotient + Remainder
Question: Verify the division of x² + 3x + 2 by x + 1 when the quotient is x + 2 and the remainder is 0.
Calculation:
Step 1: Divisor × Quotient = (x + 1)(x + 2).
Step 2: (x + 1)(x + 2) = x² + 3x + 2.
Step 3: Add the remainder: x² + 3x + 2 + 0 = x² + 3x + 2.
Answer: The identity is verified, so the quotient is x + 2 and the remainder is 0.
Statement: When a polynomial f(x) is divided by x − a, the remainder is the value f(a).
Remainder = f(a) when f(x) is divided by (x − a)
Question: Find the remainder when f(x) = x³ − 4x + 1 is divided by x − 2.
Calculation:
Step 1: For divisor x − 2, a = 2.
Step 2: Remainder = f(2).
Step 3: f(2) = 2³ − 4(2) + 1 = 8 − 8 + 1 = 1.
Answer: The remainder is 1.
Statement: The expression x − a is a factor of polynomial f(x) if and only if substituting x = a gives zero.
(x − a) is a factor of f(x) ⇔ f(a) = 0
Question: Show that x − 2 is a factor of f(x) = x² − 5x + 6.
Calculation:
Step 1: For x − 2, take a = 2.
Step 2: f(2) = 2² − 5(2) + 6.
Step 3: f(2) = 4 − 10 + 6 = 0.
Answer: Since f(2) = 0, x − 2 is a factor of f(x).
