ALGEBRA EQUATIONS - SOLVING POLYNOMIAL EQUATIONS OF HIGHER DEGREE WITH EXAMPLES

Please study introduction given to equations and Algebra Equations in Math Equations, if you have not already done so. We studied about equations of first degree in one and two variables and word problems on them in Linear Equations. We also covered equations of second degree, their solution, nature of roots, relationship between roots and coefficients, word problems in Quadratic Equations. In
Polynomials, we covered definition, degree, zeros of polynomials, remainder theorem, factor theorem. Applications of factor theorem and Synthetic Division to find the factors of polynomials were covered in Algebra Factoring. Please study all these before proceeding further. They are prerequisites here.

Here, we study Polynomial Equations of higher degree. In Algebra Equations, we cover Polynomial Equations.

Fundamental Theorem of Algebra : Algebra Equations

Every polynomial equation f(x) = 0 of degree n ≥ 1 has atleast one rootreal or complex.

Based on the Fundamental Theorem of Algebra and mathematical induction, we can prove that

Every Polynomial equation of degree n ≥ 1 has only n roots real or imaginary and no more.

This is an important point in Algebra Equations.

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Relation between the roots and coefficients of an nth degree equation : Algebra Equations

Consider the nth degree equation x^{n} + p_{1}x^{(n - 1)} + p_{2}x^{(n - 2)} + p_{3}x^{(n - 3)} + .......... + p_{n} = 0 Let α_{1}, α_{2}, α_{3},α_{4},..........α_{n} be the roots of the equation. Then, s_{1} = sum of the roots = Σ α_{1} = (-1)^{1}p_{1} s_{2} = sum of the products of the roots taken two at a time = Σ α_{1}α_{2} = (-1)^{2}p_{2} s_{3} = sum of the products of the roots taken three at a time = Σ α_{1}α_{2}α_{3} = (-1)^{3}p_{3} s_{4} = sum of the products of the roots taken four at a time = Σ α_{1}α_{2}α_{3}α_{4} = (-1)^{4}p_{4} ................................................... .................................................... s_{n} = sum of the products of the roots taken all at a time = α_{1}α_{2}α_{3}α._{4}..........α_{n} = (-1)^{n}p_{n}

NOTE 1. The above relations are true only when the coefficient of x^{n} is unity. In case it is not unity, the algebra equation shouldbe divided with the coefficient of x^{n} to make it unitybefore writing the above relations between the roots and coefficients.

2. The number of relations we have is same as the degree of the equation.So there are n relations to find n roots. Though, n relations are sufficientto find n unknowns, it is practically difficult to find the roots with thehelp of the relations only.

Usually, we are given something more about the roots. If that additional data is not given, we can solve some Algebra Equations by trial and error, using remainder theorem or factor theorem, as we did in
Algebra Factoring.

3. For n = 3, we get a third degree or cubic algebra equation x^{3} + p_{1}x^{2} + p_{2}x + p_{3} = 0 Then s_{1} = α_{1} + α_{2} + α_{3} = -p_{1} s_{2} = α_{1}α_{2} + α_{1}α_{3} + α_{2}α_{3} = p_{2} s_{3} = -p_{3}

4. For n = 4, we get a fourth degree or biquadratic algebra equation x^{4} + p_{1}x^{3} + p_{2}x^{2} + p_{3}x + p_{4} = 0 Then s_{1} = α_{1} + α_{2} + α_{3} + α_{4} = -p_{1} s_{2} = α_{1}α_{2} + α_{1}α_{3} + α_{1}α_{4} + α_{2}α_{3} + α_{2}α_{4} + α_{3}α_{4} = p_{2} s_{3} = α_{1}α_{2}α_{3} + α_{1}α_{2}α_{4} + α_{1}α_{3}α_{4} + α_{2}α_{3}α_{4} = -p_{3} s_{4} = α_{1}α_{2}α_{3}α_{4} = p_{4}

Revision of knowledge of Progressions :

In solving Algebra Equations, the following knowledge is useful.

Arithmetic Progression (A.P.) : Quantities are said to be in A.P., ifthe difference of any term and its preceding term is a constant. Harmonic Progression (H.P.) : Quantities are said to be in H.P., iftheir reciprocals are in A.P. Geometric Progression (G.P.) : Quantities are said to be in G.P., ifthe ratio of any term and its preceding term is a constant.

For a cubic equation, when the roots are (i) in A.P., they are taken as α - β, α, α + β (ii) in H.P., they are taken as 1⁄(α - β), 1⁄α, 1⁄(α + β) (iii) in G.P., they are taken as α⁄β, α, αβ

For a biquadratic equation, when the roots are (i) in A.P., they are taken as α - 3β, α - β, α + β, α + 3β (ii) in H.P., they are taken as 1⁄(α - 3β), 1⁄(α - β), 1⁄α, 1⁄(α + 3β) (iii) in G.P., they are taken as α⁄β^{3}, α⁄β, αβ, αβ^{3}

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