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Factorisation by Factor Theorem and Remainder Theorem


Important Concepts

  1. The method of finding the remainder without actually performing the process of division is called Remainder Theorem.
  2. 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).
  3. When a polynomial p(x) is divided by (x + a), the remainder is the same as p(-a).
  4. 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).
  5. 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).
  6. 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.
  7. (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 - Mathematics Class 10 ICSE

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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