addition of geometric progression

bhuvaneshnick

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Example 1.6 Solve the equation x4 - 2x3 - 21x2 + 22x + 40 = 0 whose roots are in A.P.

Solution Let the roots be a - 3d, a - d, a + d, and a + 3d, so the roots equal 4a = 2. Then a = 2.

Also, the product of the roots is (a2 - 9d 2)(a2 - d 2) = 40, so:

. . . . .\(\displaystyle \left(\dfrac{1}{4}\, -\, 9d^2\right)\left(\dfrac{1}{4}\, -\, d^2\right)\, =\, 40\) or \(\displaystyle 144d^4\, -\, 40d^2 \, -\, 639\, =\, 0\)

Then d 2 = 9/4 or -7/36. Thus, \(\displaystyle d\, =\, \pm 3/2;\) the other value is inadmissible. Hence, the required roots are -4, -1, 2, 5.

my question: how the addition of geometric series gives 2 and product gives 40
 
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Example 1.6 Solve the equation x4 - 2x3 - 21x2 + 22x + 40 = 0 whose roots are in A.P.

Solution Let the roots be a - 3d, a - d, a + d, and a + 3d, so the roots equal 4a = 2. Then a = 2.

Also, the product of the roots is (a2 - 9d 2)(a2 - d 2) = 40, so:

. . . . .\(\displaystyle \left(\dfrac{1}{4}\, -\, 9d^2\right)\left(\dfrac{1}{4}\, -\, d^2\right)\, =\, 40\) or \(\displaystyle 144d^4\, -\, 40d^2 \, -\, 639\, =\, 0\)

Then d 2 = 9/4 or -7/36. Thus, \(\displaystyle d\, =\, \pm 3/2;\) the other value is inadmissible. Hence, the required roots are -4, -1, 2, 5.

my question: how the addition of geometric series gives 2 and product gives 40

The series in question is

NOT a geometric series

It is an ARITHMETIC sequence.

Does that answer your doubt?
 
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The fact that the roots are in "geometric progression" or "arithmetic progression" is not relevant. Any polynomial \(\displaystyle x^n\,+\, a_{n-2}x^{n-1}\,+\, \cdot\cdot\cdot\,+ \,a_1x\,+\, a_0\) can be factored over the complex numbers as \(\displaystyle (x\,-\, p_1)(x\,-\, p_2)\cdot\cdot\cdot(x\,- \,p_n)\) where the \(\displaystyle p_i\) are the complex zeros of the polynomial. Multiplying that product out will show you that the coefficient of \(\displaystyle x^{n-1}\) is the negative of \(\displaystyle p_1\,+\, p_2\,+ \,\cdot\cdot\cdot\,+\, p_n\), the sum of the zeros, and the constant term is plus or minus (if n is even or odd) \(\displaystyle p_1p_2\cdot\cdot\cdot p_n\), the product of the zeros. In this particular fourth degree polynomial the coefficient of \(\displaystyle x^3\) is -2, so the sum of the zeros is 2, and the constant term is 40 (and the degree, 4, is even) so the product of the zeros is 40.
 
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