We will discuss here about the polynomial equation and its roots.
If f(x) is a polynomial in x of degree ≥ 1 whose coefficients are real or complex numbers then f(x) = 0 is called its corresponding polynomial equation.
Examples of polynomial equation:
(i) 5x\(^{2}\) + 2 x  7 is a quadratic polynomial and 5x\(^{2}\) + 2 x  7 = 0 is its corresponding quadratic equation.
(ii) 2x\(^{3}\) + x\(^{2}\) + 5x  3 is a cubic polynomial and 2x\(^{3}\) + x\(^{2}\) + 5x  3 = 0 is its corresponding cubic equation.
(iii) x\(^{4}\) + x\(^{2}\)  2x + 6 is a cubic polynomial and x\(^{4}\) + x\(^{2}\)  2x + 6 = 0 is its corresponding cubic equation.
(iv) x\(^{5}\) + 2x\(^{4}\) + 2x\(^{3}\) + 4x\(^{2}\) + x + 2 is a cubic polynomial and x\(^{5}\) + 2x\(^{4}\) + 2x\(^{3}\) + 4x\(^{2}\) + x + = 0 is its corresponding equation.
If α be a value of x for which f(x) becomes zero, i.e., f(α) = 0, then α is said to be a root of the equation f(x) n= 0.
In other words,
α is called a root of the polynomial equation f(x) = 0 if f(α) = 0.
Examples of root of the polynomial equation:
(i) Let f(x) = 4x\(^{3}\) + 12x\(^{2}\)  4x  12. As 4(1)\(^{3}\) + 12(1)\(^{2}\)  4(1)  12 = 4 + 12  4  12= 0, i.e., f(1) = 0, f(x) = 0 has a root x = 1.
(ii) Let f(x) = x\(^{2}\)  2x  3. As (1)\(^{2}\)  2(1)  3 = 1 + 2  3 = 0, i.e., f(1) = 0, f(x) = 0 has a root x = 1
(iii) Let f(x) = x\(^{4}\) + x\(^{3}\) – 2x\(^{2}\) + 4x  24. As (2)\(^{4}\) + (2)\(^{3}\)  2(2)\(^{2}\) + 4(2)  24 = 16 + 8 – 8 +8 + 8 = 0, i.e., f(2) =0, f(x) has a root x = 2
(iv) Let f(x) = x\(^{3}\) + x\(^{2}\)  x  1. As (1)\(^{3}\) + (1)\(^{2}\) – (1) – 1 = 1 + 1  1  1 = 0, i.e., f(1) = 0, f(x) = 0 has a root x = 1.
`● Factorization
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