Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 4 Theory of Equations to solve questions creatively.

## Intermediate 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_{2}x^{n – 2} + ……. + a_{n} is called a polynomial in x of degree n.

→ f(x) = a_{0}x^{n} + a_{1}x^{n – 1} + a_{2}x^{n – 2} + ……. + a_{n} = 0 is called a polynomial equation in x of degree n (a_{0} ≠ 0). Every non-constant polynomial equation has atleast one root.

→ If f(α) = 0 then α is called a root of the equation f(x) = 0.

→ If f(α) = 0 then (x – α) is a factor of f(x).

**Relation between roots and coefficients of an equation:**

→ If α β γ are the roots of x^{3} + p_{1}x^{2} + p_{2}x + p_{3} = 0 then sum of the roots s_{1} = α + β + γ = – p_{1}.

Sum of the products of two roots taken at a time s_{2} = αβ + βγ + γα = p_{2}.

Product of all the roots, s_{3} = αβγ = – p_{3}.

→ If α, β, γ, δ are the roots of x^{4} + p_{1}x^{3} + p_{2}x^{2} + p_{3}x + p_{4} = 0 then sum of the roots s_{1} = α + β + γ + δ = – p_{1}.

Sum of the products of roots taken two at a time

s_{2} = αβ + αγ + αδ + βγ + βδ + γδ = p_{2}.

Sum of the products of roots taken three at a time .

s_{3} = αβγ + βγδ + γδα + δαβ = – p_{3}.

Product of the roots, s_{4} = αβγδ = 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 \(\frac{a}{r}\), a, ar.
- In H.P., then they are taken as \(\frac{1}{a-d}, \frac{1}{a^{\prime}}, \frac{1}{a+d}\).

→ For a bi quadratic equation, if the roots are

- In A.P., then they are taken as a – 3d, a – d, a + d, a + 3d.
- In C.P., then they are taken as \(\frac{1}{a-3 d}, \frac{1}{a-d}, \frac{1}{a+d}, \frac{1}{a+3 d}\).

→ In an equation with real coefficients, imaginary roots occur in conjugate pairs.

→ In an equation with rational coefficients, irrational roots occur in pairs of conjugate surds.

→ The equation whose roots are those of the equation f(x) = 0 with contrary signs is f(- x) = 0.

→ The equation whose roots are multiplied by kfa 0) of those of the proposed equation f(x) = 0 is f \(\left(\frac{x}{k}\right)\) = 0.

→ The equation whose roots are reciprocals of the roots of f(x) = 0 is f \(\left(\frac{1}{x}\right)\) = 0.

→ The equation whose roots are exceed by h than those of f(x) = 0 is f(x – h) = 0.

→ The equation whose roots are diminished by h than those of f(x) 0 is f(x + h) = 0.

→ The equation whose roots are the square of the roots of f(x) = 0 is obtained by eliminating square root from f(√x) = 0.

→ If f(x) = p_{0}x^{n} + p_{1}x^{n – 1} + p_{2}x^{n – 2} + ……. + p_{n} = 0 then to eliminate the second term,

f(x) = 0 can be transformed to f(x + h) = 0 where h = \(\frac{-p_{1}}{n \cdot p_{0}}\).

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

→ A reciprocal equation f(x) = p_{0}x^{n} + p_{1}x^{n – 1} + …… + p^{n} = 0 is said to be a reciprocal equation of first class p_{i} = p_{n – i} for all i.

→ A reciprocal equation f(x) = p_{0}x^{n} + p_{1}x^{n – 1} + …… + p^{n} = 0 is said to be a reciprocal equation of second class if p_{i} = – p_{n – i} for all i.

→ For an odd degree reciprocal equation of class one, – 1 is a root and for an odd degree reciprocal equation of class two, 1 is a root.

→ For an even degree reciprocal equation of class two, 1 and – 1 are roots.

→ If f(x) = 0 is an equation of degree ‘n’ then to eliminate r^{th} term, f(x) = 0 can be transformed to f(x + h) = 0 where h is a constant such that f^{(n – r + 1)}(h) = 0 i.e.,(n – r + 1)^{th} derivative of f(h) is zero.

→ Every n^{th} degree equation has exactly n roots real or imaginary.

→ Relation between, roots and coefficients of an equation.

(i) If α, β, γ are the roots of x^{3} + p_{1}x^{2} + p_{2}x + p_{3} = 0 the sum of the roots s_{1} = α + β + γ = -p_{1}.

Sum of the products of two roots taken at a time s_{2} = αβ + βγ + γα = -p_{2}

Product of all the roots, s_{3} = αβγ= – p_{3}.

(ii) If α, β, γ, δ are the roots of x^{4} + p_{1}x^{3} + p_{2}x^{2} + p_{3}x + p_{4} = 0 then

- Sum of the roots s
_{1}= a+P+y+S = -p_{1}.

s_{2}= αβ + αγ + αδ + βα + βδ + γδ = p_{2}. - Sum of the products of roots taken three at a time

s_{3}= αβγ + βγδ + γδα + δαβ = – p_{3}. - Product of the roots, s
_{4}= αβγδ = p_{4}

→ For the equation x^{n} + p_{1}x^{n-1} + p_{2}x^{n-2} + ……… + p_{n} = 0

- Σ α
_{2}= p_{1}^{2}– 2p_{2} - Σ α
_{3}= -p_{1}^{3}+ 3p_{1}p_{2}– 3p_{3} - Σ α
_{4}=p_{1}^{4}– 4p_{1}^{2}p^{2}+ 2p_{2}^{2}+ 4p_{1}p_{3}– 4p_{4} - Σ α
_{2}β = 3p_{3}– p_{1}p_{2} - Σ α
_{2}βγ = p_{1}p_{3}– 4p_{4}

Note: For the equation x^{3} + p_{1}x^{2} + p_{2}x + p_{3} = 0 Σα_{2}β_{2} — p_{2} -2p_{1}p_{3}

→ To remove the second term from a nth degree equation, the roots must be diminished by \(\frac{-\mathrm{a}_{1}}{\mathrm{na}_{0}}\) and the resultant equation will not contain the term with x^{n-1}.

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

- Whose roots are \(\) is f\(\left(\frac{1}{x}\right)\) = 0
- Whose roots are kα
_{1}, kα_{2}…,kα_{n}is f\(\left(\frac{x}{h}\right)\) = 0 - Whose roots are α
_{1}– h, α_{2}– h, …. α_{n}– h is f(x + h) = 0. - Whose roots are α
_{1}+ h, α_{2}+ h, ………….. α_{n}+ h is f(x – h) = 0 - Whose roots are α
_{1}^{2}, α_{2}^{2}…. α_{1}^{2}is f (f√y) = 0

→ In any equation with rational coefficients, irrational roots occur in conjugate pairs.

→ In any equation with real coefficients, complex roots occur in conjugate pairs.

→ If α is r – multiple root of f(x) = 0, then a is a (r – 1) – multiple root of f^{1}(x) = 0 and (r-2) – Multiple root of f ^{11}(x) = 0 and non multiple root of f^{r-1}(x) =0.

→ If f(x) = x^{n} + p_{1}x^{n-1} + …………. + p_{n-1}x + p_{n} and f(a) and f(b) are of opposite sign, then at least

one real root of f(x) =0 lies between a and b.

(a) 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 are taken as \(\frac{a}{r}\), a, ar
- In H.P., then they are taken as \(\frac{1}{a-d}, \frac{1}{a}, \frac{1}{a+d}\)

(b) For a bi quadratic equation, if the roots are

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

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

- A reciprocal equation f (x) = p
_{0}x^{n}+ p_{1}x^{n-1}+ ……………….. + p_{n}= 0 is said to be a reciprocal equation of first class p_{i}= p_{n-i}for all i. - A reciprocal equation f (x) = p
_{0}x^{n}+ p_{1}x^{n-1}+ ……………….. + p_{n}= 0 = 0 is said to be a reciprocal equation of second class p_{i}= p_{n-i}for all i. - For an odd degree reciprocal equation of class one, -1 is a root and for an odd degree reciprocal equation of class two, 1 is a root.
- For an even degree reciprocal equation of class two, 1 and -1 are roots.

→ If f(x) = 0 is an equation of degree ‘n’ then to eliminate rth term, .f(x) = 0 can be transformed to f(x+h) = 0 where h is a constant such that f^{(n-r+1)}(h) =0 i.e., (n – r + 1)^{th} derivative of f(h) is zero.