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    Roots of Polynomial Equations-1

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    Shivam Nadimpalli
    There are three key points about polynomial equations: 1) Polynomials have roots equal to the number of powers of x, with roots possibly being repeated or complex. The polynomial can be reconstructed from its roots. 2) For quadratic equations, the roots' sum and product are related to the equation's coefficients. Symmetrical functions of roots can be written in terms of the sum and product. 3) For cubic equations, the roots' sum, sum of products, and product are related to the coefficients. Symmetrical functions are written in terms of the sum, sum of products, and product of roots. Relationship between roots provides information if they are in arithmetic or geometric progression.

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    Roots of Polynomial Equations-1

    Uploaded by

    Shivam Nadimpalli
    152 views2 pages
    There are three key points about polynomial equations: 1) Polynomials have roots equal to the number of powers of x, with roots possibly being repeated or complex. The polynomial can be reconstructed from its roots. 2) For quadratic equations, the roots' sum and product are related to the equation's coefficients. Symmetrical functions of roots can be written in terms of the sum and product. 3) For cubic equations, the roots' sum, sum of products, and product are related to the coefficients. Symmetrical functions are written in terms of the sum, sum of products, and product of roots. Relationship between roots provides information if they are in arithmetic or geometric progression.

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    Finding the roots and solutions of polynomial equations.

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    There are three key points about polynomial equations: 1) Polynomials have roots equal to the number of powers of x, with roots possibly being repeated or complex. The polynomial can be reconstructed from its roots. 2) For quadratic equations, the roots' sum and product are related to the equation's coefficients. Symmetrical functions of roots can be written in terms of the sum and product. 3) For cubic equations, the roots' sum, sum of products, and product are related to the coefficients. Symmetrical functions are written in terms of the sum, sum of products, and product of roots. Relationship between roots provides information if they are in arithmetic or geometric progression.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOC, PDF, TXT or read online from Scribd
    Download as doc, pdf, or txt
    152 views2 pages

    Roots of Polynomial Equations-1

    Uploaded by

    Shivam Nadimpalli
    There are three key points about polynomial equations: 1) Polynomials have roots equal to the number of powers of x, with roots possibly being repeated or complex. The polynomial can be reconstructed from its roots. 2) For quadratic equations, the roots' sum and product are related to the equation's coefficients. Symmetrical functions of roots can be written in terms of the sum and product. 3) For cubic equations, the roots' sum, sum of products, and product are related to the coefficients. Symmetrical functions are written in terms of the sum, sum of products, and product of roots. Relationship between roots provides information if they are in arithmetic or geometric progression.

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    Download as DOC, PDF, TXT or read online from Scribd
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    Roots of Polynomial Equations

    Basics Polynomials have one root for each power of x. Quadratics have 2 roots, cubics 3 roots etc. The roots can be identical (repeated roots) and do not have to be real (if there are complex roots then they come in complex conjugate pairs). The polynomial equation can be reconstructed from the roots by multiplying factors: (x )(x ) = 0 Quadratic Equations By co-efficients ax2 + bx + c = 0
    b c x2 + x + = 0 a a
    2

    By factors (x )(x )= 0 x ( + )x + = 0
    c a

    b Equating co-efficients shows that: + = a ;

    Symmetrical Functions We can write many functions of and . Those which are unchanged by swapping and are said to be symmetrical. Examples;
    + ; ; 2 + 2 ;
    1 1 +

    All symmetrical functions can be written in terms of the two basic functions: + ; . For Example; 2 + 2 = ( + ) 2 2
    1 1 + + =

    3 + 3 = ( + ) 3 3 ( + )

    Creating new equations We can create equations with roots related to the original equation. There are two ways to do this (a) by working out the new values of + and and (b) by directly developing the new equation. Example: If the roots of x2+3x+10 = 0 are and , find the equation whose roots are 3 and 3 By calculation + = -3; = 10 3 + 3 = 3( + ) = -9; (3)(3) = 9 = 90 So new equation is x2 + 9x + 90 = 0 Directly Let u = 3 Then = u/3 but satisfies x2+3x+10 = 0 2 So + 3 +10 =0
    u 2 3u + + 10 = 0 9 3

    u2 + 9u + 90 =0

    Cubic Equations The cubic equation ax3 + bx2 + cx + d = 0 has roots , and . Equating co-efficients shows that: + +
    b c d = ; + + = ; = a a a

    Symmetrical functions must be unchanged by the interchange of any two of , and . So the following are NOT symmetrical:
    + ; 2 + 2 + 2;

    All symmetrical functions can be written in terms of the three basic functions:
    + + ; + + ;

    Relationship between roots If roots in arithmetic progression then use -d, , +d. Hence the sum of roots is just 3 and thus one root is equal to b/3a If the roots are in geometric progression use /r,,r. Hence the product of roots is equal to 3 and then thus one of the roots is equal to
    3

    d . a

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