Polynomial: An algebraic expression having the exponent of the variable as non negative integers is known as a polynomial.
E.g. p(x) = ax²+bx+c, a ≠ 0, where a, b, c are constants and x is the variable.
Degree of a polynomial: In a polynomial the highest power of the variable is known as the degree of that polynomial.
In the above polynomial the highest power of the variable is 2. Therefore degree of that polynomial is 2.
Relation between zeroes and coefficients of a quadratic polynomial:
Let 𝜶 and 𝜷 be the zeroes of the quadratic polynomial p(x) = ax² + bx + c, a ≠ 0, then
(i) Sum of the zeroes = 𝜶 + 𝜷 = — b/a
(ii) Product of the zeroes = 𝜶𝜷 = c/a
To form a quadratic polynomial when the zeroes are given 𝜶 and 𝜷:
Required polynomial is
p(x) = k[x² — (𝜶 + 𝜷)x + 𝜶𝜷], where k ia an arbitrary non-zero constant.
Relation between zeroes and coefficients of a cubic polynomial:
Let 𝜶, 𝜷 and 𝜸 be the zeroes of the cubic polynomial p(x) = ax³ + bx² + cx + d, a ≠ 0, then
(i) Sum of the zeroes = 𝜶 + 𝜷 + 𝜸 = —b/a
(ii) Sum of product of the zeroes taken two at a time = 𝜶𝜷 + 𝜷𝜸 + 𝜶𝜸 = c/a
(iii) Products of the zeroes = 𝜶𝜷𝜸 = —d/a
To form a cubic polynomial when the zeroes are given 𝜶, 𝜷 and 𝜸:
Required polynomial is
p(x) = k[x³ — (𝜶 + 𝜷 +𝜸)x² + (𝜶𝜷 + 𝜷𝜸 + 𝜸𝜶)x — 𝜶𝜷𝜸 ], where k ia an arbitrary non-zero constant.
Note: Given a polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at atmost n points. Therefore, a polynomial p(x) of degree n has at most n zeroes.
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