Polynomials {Class-10 / CBSE}

Polynomial - 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)}