Learning Objectives
- Understand the geometrical meaning of zeroes of a polynomial
- Find the relationship between zeroes and coefficients of a polynomial
- Perform division algorithm for polynomials
- Find zeroes of quadratic polynomials and verify the relationships
Key Concepts
Zeroes of a Polynomial
A zero of a polynomial p(x) is a value of x for which p(x) = 0. Geometrically, the zeroes are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
- A linear polynomial ax + b has exactly one zero: x = -b/a
- A quadratic polynomial ax² + bx + c has at most two zeroes
- A cubic polynomial has at most three zeroes
Relationship Between Zeroes and Coefficients
For a quadratic polynomial ax² + bx + c with zeroes α and β:
- Sum of zeroes: α + β = -b/a
- Product of zeroes: αβ = c/a
For a cubic polynomial ax³ + bx² + cx + d with zeroes α, β, γ:
- α + β + γ = -b/a
- αβ + βγ + αγ = c/a
- αβγ = -d/a
Division Algorithm for Polynomials
If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that:
p(x) = g(x) × q(x) + r(x)
where r(x) = 0 or degree of r(x) < degree of g(x).
Summary
Polynomials are algebraic expressions with non-negative integer exponents. The zeroes of a polynomial correspond to the x-intercepts of its graph. There are definite relationships between the zeroes and coefficients. The division algorithm for polynomials is analogous to Euclid's division lemma for integers.
Important Terms
- Polynomial
- An expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ with non-negative integer exponents
- Zero of a Polynomial
- A value of x that makes the polynomial equal to zero
- Degree
- The highest power of the variable in the polynomial
- Quadratic Polynomial
- A polynomial of degree 2, of the form ax² + bx + c where a ≠ 0
Quick Revision
- For ax² + bx + c: sum of zeroes = -b/a, product of zeroes = c/a
- A quadratic can have 0, 1, or 2 real zeroes depending on discriminant
- If α and β are zeroes, the polynomial is k[x² - (α+β)x + αβ]
- Division algorithm: Dividend = Divisor × Quotient + Remainder
- Geometrically, zeroes are x-intercepts of the polynomial graph