CBSE Class 10 Mathematics Polynomials Notes Set 11

Polynomials: Zeros and Graphs

A polynomial in x is an expression with non-negative whole-number powers of x. Its degree is the highest power: degree 1 is linear, degree 2 quadratic and degree 3 cubic. A real number a is a zero of p(x) when p(a) = 0; geometrically it is a point where the graph meets the x-axis.
A linear graph is a straight line meeting the x-axis once. A quadratic graph is a parabola that opens upwards if a > 0 and downwards if a < 0, with vertex \( \left(-\frac{b}{2a},\ -\frac{D}{4a}\right) \), where \( D=b^2-4ac \). If D > 0 it cuts the x-axis twice, if D = 0 it just touches it, and if D < 0 it does not meet it. A cubic graph meets the x-axis at least once and at most three times.

Zeros and Coefficients

For \( ax^2+bx+c \) with zeros \( \alpha,\beta \): \( \alpha+\beta=-\frac{b}{a} \) and \( \alpha\beta=\frac{c}{a} \). A quadratic with given zeros is \( k\{x^2-(\alpha+\beta)x+\alpha\beta\} \), k any non-zero number.
For \( ax^3+bx^2+cx+d \) with zeros \( \alpha,\beta,\gamma \): \( \alpha+\beta+\gamma=-\frac{b}{a} \), \( \alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a} \) and \( \alpha\beta\gamma=-\frac{d}{a} \). Useful identity: \( \alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta \).

Remainder and Factor Theorems

Remainder theorem: dividing p(x) by (x - a) leaves remainder p(a). For a divisor such as 1 - 2x, set it to zero and evaluate at x = 1/2.
Factor theorem: (x - a) divides p(x) exactly when p(a) = 0. To factorise a higher-degree polynomial, try small factors of the constant term divided by factors of the leading coefficient, then divide out each factor found.

Division Algorithm

For polynomials p(x) and g(x) with g not zero, p(x) = g(x) q(x) + r(x), where r(x) is zero or has degree less than g(x). To find what must be added to (or subtracted from) p(x) to make it divisible by g(x), divide and use the remainder: subtract the remainder from p(x), that is, add its negative.
Worked example: for p(x) = x^3 - 6x^2 - 15x + 80 and g(x) = x^2 + x - 12 the quotient is x - 7 and the remainder is 4x - 4, so 4x - 4 must be subtracted for exact divisibility. Another: if the sum of zeros of a quadratic is 5 and the product is 17, the polynomial is k(x^2 - 5x + 17).