\( \DeclareMathOperator{cosec}{cosec} \)

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The factor theorem and remainder theorem are fundamental concepts in polynomial algebra. The factor theorem states that for a polynomial \( P(x) \), if \( P(a) = 0 \), then \( (x-a) \) is a factor of \( P(x) \). The remainder theorem, on the other hand, states that when a polynomial \( P(x) \) is divided by a linear divisor \( x-a \), the remainder is \( P(a) \).

Key Formulae:

$$ P(a) = 0 \implies (x-a) \text{ is a factor of } P(x) $$ $$ \text{Remainder of } P(x) \text{ when divided by } (x-a) = P(a) $$Example:

Consider the polynomial \( P(x) = x^2 - 3x + 2 \). Using the factor theorem, if \( P(1) = 0 \), then \( (x-1) \) is a factor of \( P(x) \).

$$ P(1) = 1^2 - 3(1) + 2 = 0 $$Thus, \( (x-1) \) is a factor of \( P(x) \). Additionally, using the remainder theorem, the remainder when \( P(x) \) is divided by \( x-1 \) is \( P(1) \), which is 0 in this case.

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