0Polynomial - Concept
A polynomial is a mathematical expression consisting of the sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient.
Zeroes of a Polynomial.
- Mathematically: K is said to be zero of a polynomial p(x) if p(k) = 0
- Graphically: Values of x coordinates where Polynomial P(x) crosses/touches the x-axis are called zeros of polynomial P(x).
Some Graphs:
- Graph of a linear polynomial ax + b is a straight line.
- Graph of a quadratic polynomial p(x) = ax2+ bx + c is a parabola open upwards like ∪, if a > 0.
- Graph of a quadratic polynomial p(x) = ax2+ bx + c is a parabola open downwards like ∩, if a > 0.
- In general, a polynomial p(x) of degree n crosses the x-axis at at-most n points.
The relationship between zeroes and the coefficients of a Quadratic Polynomial :
- If α, β are zeroes P(x) = ax2 + bx + c, then
- If α, β are roots of a quadratic polynomial p(x), then.
P(x) =K {x2 – (sum of roots) x + product of roots}
P(x) = K {x2 + (α+ β) x +(αβ)}
The relationship between the zeroes and the coefficients of a Cubic Polynomial
If α, β and γ are zeroes of a cubic polynomial p(x),
P(x) = K {x3 – (sum of zeroes) x2 + (sum of product of zeroes) x – (product of zeroes)}
P(x) = K {x3 – (α + β + γ) x2 + (αβ + βγ + αγ) x – (αβγ)}
