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

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 ……………. a_n are real or complex numbers and a₀ ≠ 0, then the expression f(x) = a₀xⁿ + a₁x^n-1 + a₀x^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 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,. …………………….., α_n are the roots of the equation f(x) = 0, then α₁ – h, α₂ – h, ……………. , α_n – h are the roots of the equation f(x + h) = 0 and α₁ + h, α₂ + h, ………………. , α_n + h are the roots of the equation f(x – h) = 0.

→ If f(x) = a₀xⁿ + a₁x^n-1 + ………….. + aα_n = 0, then the transformed equation whose roots are the reciprocals of the roots of f(x) = 0 is Φ(x) = a₀ + a₁x + a₂x² + + a_nxⁿ = 0.

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