Learning these TS Inter 2nd Year Maths 2A Formulas Chapter 4 Theory of Equations will help students to solve mathematical problems quickly.

## TS Inter 2nd Year Maths 2A Theory of Equations Formulas

→ If n is a non-negative integer and a_{0}, a_{1}, a_{2} ……………. a_{n} are real or complex numbers and a_{0} ≠ 0, then the expression f(x) = a_{0}x^{n} + a_{1}x^{n-1} + a_{0}x^{n-2} + …………. + a_{n} is called a polynomial in x of degree n.

→ If f(x) is a polynomial of degree n > 0, then the equation f(x) = 0 is called an algebraic equation or polynomial equation of degree n.

→ A complex number a. is said to be zero of a polynomial f(x) or a root of the equation f(x) = 0, if f(α) = 0.

→ Every non-constant polynomial equation has atleast one root.

→ Relation between the roots and the coefficients of an equation:

(i) If α_{1}, α_{2}, α_{3} are the roots of x^{3} + p_{1}x^{2} + p_{2}x + p_{3} = 0

- s
_{1}= α_{1}+ α_{2}+ α_{3}= -p_{1} - s
_{2}= α_{1}α_{2}+ α_{2}α_{3}+ α_{3}α_{1}= p_{2} - s
_{3}= α_{1}α_{2}α_{3}= -p_{3}

(ii) If α_{1}, α_{2}, α_{3}, α_{4} are the roots of x4 + p,x‘4 + p.,x“ + p.,x + p4 = 0, then

- s
_{1}= α_{1}+ α_{2}+ α_{3}+ α_{4}= -p_{1} - s
_{2}= α_{1}α_{2}+ α_{2}α_{3}+ α_{3}α_{4}+ α_{1}α_{3}+ α_{2}α_{4}= p_{2} - s
_{3}= α_{1}α_{2}α_{3}+ α_{2}α_{3}α_{4}+ α_{3}α_{4}α_{1}+ α_{1}α_{2}α_{4}= -p_{3} - s
_{4}= α_{1}α_{2}α_{3}α_{4}= p_{4}

→ For a cubic equation, when the roots are

- in A.P., then they are taken as a – d, a, a + d.
- in G.P., then they are taken as \(\frac{a}{d}\), a, ad.
- in H.P., then they are taken as \(\frac{1}{a-d}, \frac{1}{a}, \frac{1}{a+d}\)

→ For a biquadratic equation, if the roots are

- in A.P.. then they are taken as a – 3d, a – d. a + d, a + 3d.
- in G.P., then they are taken as \(\frac{a}{d^3}, \frac{a}{d}\),ad, ad
^{3}. - in H.P., then they are taken as \(\frac{1}{a-3 d}, \frac{1}{a-d}, \frac{1}{a+d}, \frac{1}{a+3 d}\)

→ To find a root of f(x) = 0. we have to find out a value of x, for which f(x) = 0. Sometimes we can do this inspection. This method is known as trial and error method.

- For a polynomial equation with rational coefficients, irrational roots occur in pairs,
- For a polynomial equation with real coefficients, complex roots occur in pairs.

→ If α_{1}, α_{2},. …………………….., α_{n} are the roots of the equation f(x) = 0, then α_{1} – h, α_{2} – h, ……………. , α_{n} – h are the roots of the equation f(x + h) = 0 and α_{1} + h, α_{2} + h, ………………. , α_{n} + h are the roots of the equation f(x – h) = 0.

→ If f(x) = a_{0}x^{n} + a_{1}x^{n-1} + ………….. + aα_{n} = 0, then the transformed equation whose roots are the reciprocals of the roots of f(x) = 0 is Φ(x) = a_{0} + a_{1}x + a_{2}x^{2} + + a_{n}x^{n} = 0.

→ If an equation is unaltered by changing x into \(\frac{1}{n}\), then it is a reciprocal equation.