Polynomial Equation and its Roots

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

• Polynomial
• Polynomial Equation and its Roots
  
• Division Algorithm
  
• Remainder Theorem
  
• Problems on Remainder Theorem
  
• Factors of a Polynomial
  
• Worksheet on Remainder Theorem
  
• Factor Theorem
  
• Application of Factor Theorem

10th Grade Math

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