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REMAINDER THEOREM
If p(x) is any polynomial of degree greater than or equal to 1 and let a be any real number. When p(x) is divided by x– a, then the remainder is equal to p(a).
By division algorithm, we know that Dividend = Divisor × quotient + Remainder
d(x) = p(x) q(x) + r(x)
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REMARK 1. If a polynomial p(x) is divided by (x + a), the remainder is equal to the value of p(x) at x = – a i.e., p(–a)
2. If a polynomial p(x) is divided by (ax – b), the remainder is equal to the value of p(x) at
3. If a polynomial p(x) is divided by (b – ax), the remainder is equal to the value of p(x) at
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Ex. If the polynomials ax3 + 4x2 + 3x – 4 and x3 – 4x + a leave the same remainder when divided by (x – 3), find the value of a.
FAQs on Remainder Theorem of Polynomials - Polynomials, Class 9, Mathematics
| 1. What is the remainder theorem of polynomials? |  |
| 2. How do you use the remainder theorem to find the remainder of a polynomial? | |
Ans. To find the remainder of a polynomial when it is divided by a linear polynomial (x-a), substitute 'a' in the polynomial and find the value of P(a). This value will be the remainder of the division.
| 3. Can the remainder theorem be used to factorize a polynomial? | |
Ans. Yes, the remainder theorem can be used to factorize a polynomial. If a polynomial P(x) is divided by a linear polynomial (x-a) and the remainder is 0, then (x-a) is a factor of P(x). This can be used to factorize the polynomial.
| 4. What is the degree of the remainder when a polynomial is divided by a linear polynomial? | |
Ans. The degree of the remainder when a polynomial is divided by a linear polynomial is always less than the degree of the divisor. In other words, if the divisor is of degree 1, then the degree of the remainder will be 0 or less than 1.
| 5. Can the remainder theorem be used to solve equations? | |
Ans. Yes, the remainder theorem can be used to solve equations. If a polynomial P(x) is divided by a linear polynomial (x-a) and the remainder is 0, then the value of 'a' is a root of the equation P(x) = 0. This can be used to solve the equation.
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