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

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 a0, a1, a2, ……….. an are real or complex numbers and a0 ≠ 0, then the expression f(x) = a0xn + a1xn – 1 + a2xn – 2 + ……. + an is called a polynomial in x of degree n.

→ f(x) = a0xn + a1xn – 1 + a2xn – 2 + ……. + an = 0 is called a polynomial equation in x of degree n (a0 ≠ 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 x3 + p1x2 + p2x + p3 = 0 then sum of the roots s1 = α + β + γ = – p1.
Sum of the products of two roots taken at a time s2 = αβ + βγ + γα = p2.
Product of all the roots, s3 = αβγ = – p3.

→ If α, β, γ, δ are the roots of x4 + p1x3 + p2x2 + p3x + p4 = 0 then sum of the roots s1 = α + β + γ + δ = – p1.
Sum of the products of roots taken two at a time
s2 = αβ + αγ + αδ + βγ + βδ + γδ = p2.
Sum of the products of roots taken three at a time .
s3 = αβγ + βγδ + γδα + δαβ = – p3.
Product of the roots, s4 = αβγδ = 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 $$\frac{a}{r}$$, 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) = p0xn + p1xn – 1 + p2xn – 2 + ……. + pn = 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) = p0xn + p1xn – 1 + …… + pn = 0 is said to be a reciprocal equation of first class pi = pn – i for all i.

→ A reciprocal equation f(x) = p0xn + p1xn – 1 + …… + pn = 0 is said to be a reciprocal equation of second class if pi = – pn – 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.

→ Every nth 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 x3 + p1x2 + p2x + p3 = 0 the sum of the roots s1 = α + β + γ = -p1.
Sum of the products of two roots taken at a time s2 = αβ + βγ + γα = -p2
Product of all the roots, s3 = αβγ= – p3.

(ii) If α, β, γ, δ are the roots of x4 + p1x3 + p2x2 + p3x + p4 = 0 then

• Sum of the roots s1 = a+P+y+S = -p1.
s2 = αβ + αγ + αδ + βα + βδ + γδ = p2.
• Sum of the products of roots taken three at a time
s3 = αβγ + βγδ + γδα + δαβ = – p3.
• Product of the roots, s4 = αβγδ = p4

→ For the equation xn + p1xn-1 + p2xn-2 + ……… + pn = 0

• Σ α2 = p12 – 2p2
• Σ α3 = -p13 + 3p1p2 – 3p3
• Σ α4 =p14 – 4p12p2 + 2p22 + 4p1p3 – 4p4
• Σ α2β = 3p3 – p1p2
• Σ α2βγ = p1p3 – 4p4

Note: For the equation x3 + p1x2 + p2x + p3 = 0 Σα2β2 — p2 -2p1p3

→ 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 xn-1.

→ If α1 , α2 ………………. , αn are the roots of f(x) = 0, the equation

• Whose roots are  is f$$\left(\frac{1}{x}\right)$$ = 0
• Whose roots are kα1, kα2 …,kαn is f$$\left(\frac{x}{h}\right)$$ = 0
• Whose roots are α1 – h, α2 – h, …. αn – h is f(x + h) = 0.
• Whose roots are α1 + h, α2 + h, ………….. αn + h is f(x – h) = 0
• Whose roots are α12, α22…. α12 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 f1(x) = 0 and (r-2) – Multiple root of f 11(x) = 0 and non multiple root of fr-1(x) =0.

→ If f(x) = xn + p1xn-1 + …………. + pn-1x + pn 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, ad3
• In H.P., then they are taken as $$\frac{1}{a-3 d}, \frac{1}{a-d}, \frac{1}{a+d}, \frac{1}{a+3 d}$$

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

• A reciprocal equation f (x) = p0xn + p1xn-1 + ……………….. + pn = 0 is said to be a reciprocal equation of first class pi = pn-i for all i.
• A reciprocal equation f (x) = p0xn + p1xn-1 + ……………….. + pn = 0 = 0 is said to be a reciprocal equation of second class pi = pn-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.