Geometrical Meaning of the Zero Polynomial
* The x-coordinate of the point, where the graph of a polynomial intersects the x-axis is called the zero of the polynomial.
* An nth-degree polynomial intersects the x-axis at maximum of n points and therefore, has a maximum of n zeroes.
In a quadratic polynomial ax²+ bx + c,
* If a > 0, then the graph is a parabola that open upwards.
* If a < 0, then the graph is a parabola that open downwards.
Relationship between Zeros and Coefficients of a Polynomial
If α is the of a linear polynomial ax + b, then α = - \frac{b}{a} = - \frac{constant \thinspace \thinspace term }{coefficient \thinspace \thinspace of \thinspace \thinspace x}
If α and β are the two zeroes of a quadratic polynomial ax²+ bx + c, then:
* Sum of zeroes (α+β) = - \frac{b}{a} = - \frac{coefficient \thinspace of \thinspace x}{coefficient \thinspace \thinspace of \thinspace \thinspace x²}
* Product of zeroes (α.β) = \frac{c}{a} = \frac{constant \thinspace \thinspace term }{coefficient \thinspace \thinspace of \thinspace \thinspace x²}
If α,β and γ are the three zeroes of a cubic polynomial ax³+ bx² + cx + d, then:
* Sum of zeroes (α+β+γ ) = – \frac{coefficient \thinspace \thinspace of \thinspace x²}{coefficient \thinspace \thinspace of \thinspace \thinspace x³}
* Sum of products of zeroes taken two at time (α.β + β.γ + γ.α) = \frac{c}{a} = \frac{coefficient \thinspace \thinspace of \thinspace \thinspace x}{coefficient \thinspace \thinspace of \thinspace \thinspace x³}
* Product of zeroes (α.β.γ) = - \frac{d}{a} = - \frac{constant \thinspace \thinspace term }{coefficient \thinspace \thinspace of \thinspace \thinspace x³}
Division Algorithm for Polynomials
The division algorithm for polynomials states that if f(x) and ɡ(x) are the
polynomials such that degree of f(x) ≥ degree of ɡ(x) ≠ 0, then three exists
unique polynomials q(x) and r(x) such that f(x) = ɡ(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of ɡ(x).