CBSE Class 10 Mathematics Polynomials Notes Set 10

Polynomials and Their Degree

A polynomial in x is an expression made of non-negative whole-number powers of x with real coefficients. Its degree is the highest power of x that occurs. Degree 1, 2 and 3 polynomials are called linear, quadratic and cubic. A polynomial of degree n has at most n zeros.
A zero of p(x) is a number k with p(k) = 0. Geometrically it is the x-coordinate of a point where the graph of y = p(x) meets the x-axis.

Graphs of Quadratic Polynomials

The graph of y = ax² + bx + c is a parabola, opening upwards if a > 0 and downwards if a < 0. The discriminant D = b² − 4ac tells how many times it meets the x-axis: two points if D > 0, one point (touching) if D = 0 and no point if D < 0. The vertex lies at x = −b/2a.

Zeros and Coefficients

Quadratic ax² + bx + c with zeros α and β: α + β = −b/a and αβ = c/a. A quadratic with given zeros is k[x² − (α + β)x + αβ].
Cubic ax³ + bx² + cx + d with zeros α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a and αβγ = −d/a.
Useful identities: α² + β² = (α + β)² − 2αβ and α³ + β³ = (α + β)³ − 3αβ(α + β).

Division, Remainder and Factors

If p(x) is divided by g(x) the result is p(x) = g(x)q(x) + r(x) with deg r < deg g. By the remainder theorem the remainder on dividing by (x − a) is p(a), and (x − a) is a factor exactly when p(a) = 0.
Zeros in A.P.: take them as a − d, a, a + d; the middle zero a is one third of the sum of zeros. Irrational zeros of the form a + √b of a polynomial with rational coefficients come with the conjugate a − √b.

Worked Example

Find all zeros of x³ − x² − 4x + 4 if −2 is one zero.
Divide by (x + 2): the quotient is x² − 3x + 2 = (x − 1)(x − 2). So the zeros are −2, 1 and 2, and the sum 1 = −(−1)/1 and product −2 × 1 × 2 = −4 = −4/1 agree with the coefficient relations.