Class 10 Exam > Class 10 Notes > Mathematics (Maths) Class 10 > Points to Remember: Polynomials
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FAQs on Points to Remember: Polynomials - Mathematics (Maths) Class 10
| 1. What is a polynomial and how is it defined mathematically? |  |
Ans.A polynomial is a mathematical expression that consists of variables (also called indeterminates) raised to non-negative integer powers and multiplied by coefficients. It is defined in the form: \( P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where \( a_n, a_{n-1}, ..., a_0 \) are constants, \( n \) is a non-negative integer, and \( x \) is the variable.
| 2. What are the different types of polynomials based on their degree? | |
Ans.Polynomials can be classified based on their degree as follows:
- A constant polynomial has a degree of 0 (e.g., \( P(x) = 5 \)).
- A linear polynomial has a degree of 1 (e.g., \( P(x) = 2x + 3 \)).
- A quadratic polynomial has a degree of 2 (e.g., \( P(x) = x^2 + 4x + 4 \)).
- A cubic polynomial has a degree of 3 (e.g., \( P(x) = x^3 - x + 2 \)).
- Higher degree polynomials are classified similarly (e.g., quartic for degree 4, quintic for degree 5).
| 3. How do you add and subtract polynomials? | |
Ans.To add or subtract polynomials, combine like terms, which are terms that have the same variable raised to the same power. For example, to add \( P(x) = 2x^2 + 3x + 1 \) and \( Q(x) = x^2 + 4x + 2 \), you combine:
\( (2x^2 + x^2) + (3x + 4x) + (1 + 2) = 3x^2 + 7x + 3 \).
For subtraction, do the same but subtract the coefficients of like terms.
| 4. What is the factorization of polynomials and why is it important? | |
Ans.Factorization of polynomials involves expressing a polynomial as a product of its factors. This is important because it simplifies the polynomial, making it easier to find its roots or solutions, analyze its behavior, and solve equations. For example, the polynomial \( P(x) = x^2 - 5x + 6 \) can be factored as \( (x - 2)(x - 3) \).
| 5. What is the Remainder Theorem and how can it be applied to polynomials? | |
Ans.The Remainder Theorem states that if a polynomial \( P(x) \) is divided by \( (x - c) \), the remainder of this division is equal to \( P(c) \). This theorem can be applied to quickly evaluate polynomials at specific points and to determine whether \( c \) is a root of the polynomial. For example, if \( P(x) = x^3 - 4x + 6 \) and \( c = 2 \), then the remainder when dividing by \( (x - 2) \) is \( P(2) = 2^3 - 4(2) + 6 = 0 \), indicating that \( x = 2 \) is a root.
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