This chapter provides a comprehensive summary of polynomials class 9 and their properties. Here are the key topics covered:
Introduction to Polynomials
- Polynomials are algebraic expressions with variables and coefficients, involving terms with non-negative integer exponents.
- Examples: x²+2x+1, 3y³−4y+7
What is Polynomial?
- Definition: A polynomial p(x) in one variable x is an expression of the form p(x) = anx^n + an-1x^(n-1) + … + a2x^2 + a1x + a0, where a0, a1, a2, …, an are constants and an ≠ 0
- Terminology: Terms, coefficients, degree of a polynomial
- Degree of a Polynomial: Highest power of the variable (e.g., x³+2x has degree 3)
Types of Polynomials
- By number of terms:
- Monomial: Single term (e.g., 5x)
- Binomial: Two terms (e.g., x+1)
- Trinomial: Three terms (e.g., x²+x+1)
- By degree:
- Linear Polynomial: Degree 1 (e.g., 3x+2)
- Quadratic Polynomial: Degree 2 (e.g., x²−5x+6)
- Cubic Polynomial: Degree 3 (e.g., x³+4x²−3)
Zeros of Polynomials
- A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0
- Every linear polynomial has exactly one zero
- Non-zero constant polynomials have no zeros
- Every real number is a zero of the zero polynomial
Factorization Techniques
- Factor Theorem: x – a is a factor of polynomial p(x) if and only if p(a) = 0
- Remainder Theorem: When p(x) is divided by x – a, the remainder equals p(a)
- Methods for factorizing polynomial:
- Using the Factor Theorem
- Splitting the middle term (for quadratic polynomials)
- Using algebraic identities
Algebraic Identities
The chapter covers several important algebraic identities:
- (x + y)² = x² + 2xy + y²
- (x – y)² = x² – 2xy + y²
- x² – y² = (x + y)(x – y)
- (x + a)(x + b) = x² + (a + b)x + ab
- (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
- (x + y)³ = x³ + y³ + 3xy(x + y)
- (x – y)³ = x³ – y³ – 3xy(x – y)
- x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx)
These identities are useful for expanding expressions and factorizing this mathematical expression, making calculations more efficient, and solving various algebraic problems.
Important Concepts:
- Zero Polynomial: p(x)=0p(x)=0; degree is undefined.
- Non-zero Constant Polynomial: Degree is 0 (e.g., 77).
- Factorisation Techniques: Include splitting the middle term, using the Factor Theorem, and applying identities.
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