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Revision Notes: Remainder and Factor Theorems
Factorisation by Factor Theorem and Remainder Theorem
Important Concepts
- The method of finding the remainder without actually performing the process of division is called Remainder Theorem.
- Remainder Theorem states that if p(x) is any polynomial of degree > 1, and a is any number then if p(x) is divided by (x - a) then the remainder is p(a).
- When a polynomial p(x) is divided by (x + a), the remainder is the same as p(-a).
- If a polynomial p(x) over R is divided by ax + b (a ≠ 0 and a, b ∈ R) then the remainder is p(- b/a).
- Factor Theorem states that if p(x) is a polynomial of degrees > 0 then it follows from the remainder theorem that
(a) p(x) = (x - a) q(x) + p(a) Where q(x) is a polynomial of degree n - 1.
(b) If p(a) = 0 then p(x) = (x - a) q(x).
(c) Thus, if p(a) = 0, then (x - a) is a factor of p(x). - ax + b (a ≠ 0, a, b ∈ R) is a factor of the polynomial p (x) over R if and only if p ((-b)/a) = 0.
- (x - a)(x - b) is a factor of the polynomial p(x), iff p(a) = 0 and p(b) = 0.
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FAQs on Revision Notes: Remainder and Factor Theorems
| 1. What is the Remainder Theorem and how is it used in polynomial division? |  |
Ans. The Remainder Theorem states that if a polynomial \( f(x) \) is divided by a linear divisor \( (x - c) \), the remainder of that division is equal to \( f(c) \). This theorem is useful because it allows us to evaluate the polynomial at a specific point without performing long division. For example, if we want to know the remainder of \( f(x) = x^3 - 2x^2 + x - 5 \) when divided by \( (x - 3) \), we simply calculate \( f(3) \), which gives us the remainder directly.
| 2. What is the Factor Theorem and how does it relate to the Remainder Theorem? | |
Ans. The Factor Theorem is a special case of the Remainder Theorem. It states that a polynomial \( f(x) \) has a factor \( (x - c) \) if and only if \( f(c) = 0 \). This means that if we substitute \( c \) into the polynomial and get a result of zero, then \( (x - c) \) is a factor of \( f(x) \). This theorem is particularly useful for finding the factors of polynomials and for solving polynomial equations.
| 3. How can I use the Remainder Theorem to check if a number is a root of a polynomial? | |
Ans. To check if a number \( c \) is a root of a polynomial \( f(x) \), you can apply the Remainder Theorem. Simply evaluate \( f(c) \). If the result is zero, then \( c \) is a root of the polynomial, meaning \( (x - c) \) is a factor of \( f(x) \). For example, to check if \( x = 2 \) is a root of \( f(x) = x^2 - 4 \), we calculate \( f(2) = 2^2 - 4 = 0 \). Since the result is zero, \( x = 2 \) is indeed a root.
| 4. Can the Remainder Theorem be applied to polynomials of any degree? | |
Ans. Yes, the Remainder Theorem can be applied to polynomials of any degree. It is a general principle that applies regardless of whether the polynomial is of degree one, two, or higher. The key requirement is that the divisor must be a linear polynomial of the form \( (x - c) \). For instance, for a polynomial of degree 5, you can still use the Remainder Theorem to find the remainder when it is divided by \( (x - c) \) by simply evaluating the polynomial at \( c \).
| 5. How do I factor a polynomial using the Factor Theorem? | |
Ans. To factor a polynomial using the Factor Theorem, follow these steps: First, identify potential roots \( c \) by testing values, often starting with factors of the constant term. Evaluate the polynomial \( f(c) \). If \( f(c) = 0 \), then \( (x - c) \) is a factor of \( f(x) \). Once you find a factor, you can perform polynomial long division or synthetic division to divide \( f(x) \) by \( (x - c) \) and find the quotient polynomial. Continue this process until the polynomial is completely factored.
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