Relation between the roots of a polynomial equation of degree n:

Suppose the equation

            a_nxⁿ + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + .... + a₁x + a₀ = 0              . . . .  (1)

( where a₀, a₁...., a_n are real coefficients and a_n ≠ 0)

Let       α₁, α₂,....,α_n be the roots of equation (1). Then

            a_nxⁿ + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + ..... + a₁x + a­₀ º a_n(x - a₁) (x - a₂) ..... (x - a_n)

equating the coefficients of like powers of x, we have

 Some important results:

•   A polynomial relation of degree n has n roots (imaginary or real).
•   If each coefficient is real then the imaginary roots present in pairs i.e. number of complex roots is usually even.
•   If the degree of a polynomial equation is not even then the number of real roots can also be odd. It follows that at least one of the roots may be real.
•   Factor theorem: If α is a root of the equation f(x)=0, then f(x) is accurately divisible by (x- α) and equally, if f(x) is purely divisible by (x- α) then a is a root of the equation f(x)=0.
•   Suppose f(x)=0 be a polynomial equation and q and p are two real numbers, then f(x)=0 may have at least one real root or an odd number of roots in between p and q if f(p) and f(q) are of reverse sign. But if f(p) and f(q) are of similar signs, then either f(x)=0 has no real roots or an even number of roots between q and p.
•   If α is  repeated root repeating r times of a polynomial relation f(x) = 0 of  degree n i.e. f(x) = (x -  α )^r g(x) ,  where g(x) is a polynomial of  degree  n - r  and  g( α ) ≠ 0,then f(α) = f'(α) = f''(α) = . . . . =  f ^(r-1)(α) = 0  and f ^r (α) ≠ 0.

The cubic function f(x) = ax³ + bx² + cx + d, where x ∈ R

Consider a > 0, the graph of f has the subsequent properties:

•   As x → ∞ , y  → ∞ due to the x³ term is positive and may dominate the related terms when x is large.
•   As x  → ∞, y  → ∞
•   A consideration of (i) and (ii) denotes that the graph of f have to cross the x-axis at least once, taking that point x = b (say), we have
• ax³ + bx² + cx + d = (x - β)Q      where Q is a quadratic expression in x.
• The equation Q = 0 may have two real and distinct roots, two real coincident roots or no real roots.
• f'(x) = 3ax² + 2bx + c and the relation f'(x) = 0 can have two real distinct roots, no real roots or two real coincident roots.
•   If f'(x) = 0 has two real distinct roots q and p where p > q, f(p) is a minimum value of f(x) and f(q) is maximum value of f(x).
• If f'(x) = 0 has two real coincident roots, r (say) then the fixed value of f(x) at (r, f(r)) is a point of inflexion.
• If f'(x) = 0 has no real and same roots, the graph of f has no fixed points.
•  f'(x) = 6ax + 2b and f''(x) = 0, f'''(x) ≠ 0 when x = -b/2a

             That shows that all cubic function have a point of inflexion.

Email based Relation between the roots of a polynomial equation Assignment Help - Homework Help

We at www.expertsmind.com offer email based Relation between the roots of a polynomial equation assignment help - homework help and projects assistance from k-12 school level to university and college level and engineering and management studies. We provide finest service of Mathematics assignment help and Mathematic homework help. Our experts are helping students in their studies and they offer instantaneous tutoring assistance giving their best practiced knowledge and spreading their world class education services through e-Learning program.

Expertsmind's best education services

• Quality assignment-homework help assistance 24x7 hrs
• Best qualified tutor's network
• Time on delivery
• Quality assurance before delivery
• 100% originality and fresh work