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 a0, a1, a2 ……………. an are real or complex numbers and a0 ≠ 0, then the expression f(x) = a0xn + a1xn-1 + a0xn-2 + …………. + an 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 α1, α2, α3 are the roots of x3 + p1x2 + p2x + p3 = 0
- s1 = α1 + α2 + α3 = -p1
- s2 = α1α2 + α2α3 + α3α1 = p2
- s3 = α1α2α3 = -p3
(ii) If α1, α2, α3, α4 are the roots of x4 + p,x‘4 + p.,x“ + p.,x + p4 = 0, then
- s1 = α1 + α2 + α3 + α4 = -p1
- s2 = α1α2 + α2α3 + α3α4 + α1α3 + α2α4 = p2
- s3 = α1α2α3 + α2α3α4 + α3α4α1 + α1α2α4 = -p3
- s4 = α1α2α3α4 = p4
→ 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, ad3.
- 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 α1, α2,. …………………….., αn are the roots of the equation f(x) = 0, then α1 – h, α2 – h, ……………. , αn – h are the roots of the equation f(x + h) = 0 and α1 + h, α2 + h, ………………. , αn + h are the roots of the equation f(x – h) = 0.
→ If f(x) = a0xn + a1xn-1 + ………….. + aαn = 0, then the transformed equation whose roots are the reciprocals of the roots of f(x) = 0 is Φ(x) = a0 + a1x + a2x2 + + anxn = 0.
→ If an equation is unaltered by changing x into \(\frac{1}{n}\), then it is a reciprocal equation.