Polynomials — Class 10 Mathematics

A polynomial is an algebraic expression consisting of terms with non-negative integer exponents and real coefficients.

Standard form: p(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ
where aₙ ≠ 0, and n = degree of polynomial

Classification by Degree:

• Constant polynomial: Degree 0 (e.g., 5, -3)
• Linear polynomial: Degree 1 (e.g., 2x + 1)
• Quadratic polynomial: Degree 2 (e.g., x² + 3x + 2)
• Cubic polynomial: Degree 3 (e.g., x³ - 2x² + x - 1)
• Biquadratic polynomial: Degree 4 (e.g., x⁴ + x² + 1)

Key Points:

• Exponents must be non-negative integers (no √x or x⁻¹)
• Degree = highest exponent of x
• Zero polynomial has no degree
• Coefficients can be positive, negative, or zero
• A polynomial of degree n has at most n zeros

A zero (or root) of a polynomial p(x) is a value 'a' such that p(a) = 0.

Definition: 'a' is a zero of p(x) if p(a) = 0

Finding Zeros:

Key Points:

• A polynomial of degree n has at most n real zeros
• Geometrically, zeros are x-intercepts of the polynomial graph
• If 'a' is a zero, then (x - a) is a factor of p(x)

There is a direct relationship between the zeros and coefficients of a polynomial, especially for quadratic and cubic polynomials.

For a Quadratic Polynomial: ax² + bx + c with zeros α and β
Sum of zeros: α + β = -b/a
Product of zeros: α × β = c/a

For a Cubic Polynomial: ax³ + bx² + cx + d with zeros α, β, γ
Sum of zeros: α + β + γ = -b/a
Sum of products (pair): αβ + βγ + γα = c/a
Product of zeros: α × β × γ = -d/a

Key Applications:

• Finding missing zero: If two zeros are known, find the third using sum
• Forming polynomial: If zeros are given, construct the polynomial
• Verifying roots: Check if found values satisfy sum and product formulas

Forming Polynomial from Zeros:

The Division Algorithm states that when a polynomial is divided by another, we get quotient and remainder.

Division Algorithm: If p(x) and g(x) are polynomials with g(x) ≠ 0,
p(x) = g(x) × q(x) + r(x)
where q(x) is quotient, r(x) is remainder, and degree(r) < degree(g)

Terminology:

• Dividend: p(x) (the polynomial being divided)
• Divisor: g(x) (the polynomial dividing)
• Quotient: q(x) (the result)
• Remainder: r(x) (what's left over)

Steps for Polynomial Long Division:

1. Arrange both polynomials in descending order of power
2. Divide the first term of dividend by first term of divisor
3. Multiply the entire divisor by this quotient term
4. Subtract from the dividend
5. Repeat until degree of remainder < degree of divisor

Key Points:

• For linear divisor (x - a), remainder is always a constant
• Division by (x - 1) is simpler than general linear divisors
• This algorithm is essential for remainder and factor theorems

The Remainder Theorem provides a shortcut to find the remainder without full division.

Remainder Theorem: If p(x) is divided by (x - a), the remainder is p(a).

Factor Theorem: (x - a) is a factor of p(x) if and only if p(a) = 0

Practical Uses:

• Find remainder quickly: No need for long division if using remainder theorem
• Check if factor exists: Substitute the value and check if p(a) = 0
• Factorize polynomials: Find zeros, use factor theorem to identify factors

Key Points:

• Remainder theorem works for any polynomial divided by linear (x - a)
• Factor theorem is a special case: remainder = 0 means (x - a) is a factor
• If p(a) ≠ 0, then (x - a) is NOT a factor

Factorization means expressing a polynomial as a product of its factors.

Common Factorization Methods:

Strategy for Factorization:

1. Look for common factors first
2. For quadratics, use ac method or identify perfect squares
3. For cubics, use factor theorem to find one zero
4. After removing one factor, factor the remaining polynomial
5. Verify by expanding