polynomial

Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.

Zero of polynomial

A real number k is said to be a zero of a polynomial p(x), if p(k) = 0

The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x) intersects the x-axis.

Zero of a linear polynomial ax + b is - b/a

• A n-th degree polynomial can have at most n zeroes

Relationship between Zeroes and Coefficients of a Polynomial

if α* and β* are the zeroes of the quadratic polynomial p(x) = ax² + bx + c, a ≠ 0,

then you know that x – α and x – β are the factors of p(x).

Therefore, ax²+ bx + c = k(x – α) (x – β), where k is a constant

ax²+ bx + c = k[x² – (α + β)x + α β]

ax²+ bx + c = kx²– k(α + β)x + k α β

Comparing the coefficients of x², x and constant terms on both the sides, we get

a = k, b = – k(α + β) and c = kαβ.

This gives α + β = -b/a ,

αβ = c/a

Sum of zeroes = α + β = - b/a

= - (Coefficient of x) / (Coefficient of x²)

Product of zeroes = αβ = c/a

= Constant term / Coefficient of x²

• If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then

  • α + β = - b/a
  • αβ = c/a

• If α, β, γ are the zeroes of the cubic polynomial ax³+ bx² + cx + d, then

  • α + β + γ = - b/a
  • αβ + βγ + γα = c/a
  • αβγ = - d/a

A pair of linear equations in two variables can be represented, and solved, by the:

1. Graphical method

2. Algebraic method

Graphical Method

The graph of a pair of linear equations in two variables is represented by two lines.

• If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.

• If the lines coincide, then there are infinitely many solutions — each point on the line being a solution. In this case, the pair of equations is dependent (consistent).

• If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.

Algebraic Methods

We have discussed the following methods for finding the solution(s) of a pair of linear equations :

1. Substitution Method

2. Elimination Method

If a pair of linear equations is given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then the following situations can arise :

• [ a1/a2 ≠ b1/b2 ] : In this case, the pair of linear equations is consistent.

• [ a1/a2 = b1/b2 ≠ c1/c2 ] : In this case, the pair of linear equations is inconsistent.

• [ a1/a2 = b1/b2 = c1/c2 ] : In this case, the pair of linear equations is dependent and consistent.