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Solve the Polynomial Equation 4x3 - 24x2 + 23x + 18 = 0, the roots of which are in A.P.
Solution to Example 1 of Polynomial Equation : The given Algebra Equation is 4x3 - 24x2 + 23x + 18 = 0 Dividing both sides of the equation by 4, we get x3 - 6x2 + (23⁄4)x + 9⁄2 = 0 Comparing this with x3 + p1x2 + p2x + p3 = 0, we get p1 = -6; p2 = (23⁄4); p3 = 9⁄2
By data the roots are in A.P. Let the roots be α - β, α, α + β We know, sum of roots = -p1 ⇒ α - β + α + α + β = 6 ⇒ 3α = 6⇒ α = 2.........(i)
sum of the product of the roots taken two at a time = p2 = (23⁄4) ⇒ (α - β)(α) + (α - β)(α + β) + (α)(α + β) = (23⁄4) ⇒ 3α2 - β2 = (23⁄4) Using the value of α = 2 from (i), we get 3(2)2 - β2 = (23⁄4)⇒ 48 - 4β2 = 23 ⇒ 4β2 = -23 + 48 = 25 ⇒ 2β = ±5⇒ β = ±5⁄2.......(ii)
Solution to Example 2 of Polynomial Equation : The given algebra eqaution is x3 - 5x2 - 4x + 20 = 0
Here, no additional condition is given. Let f(x) = x3 - 5x2 - 4x + 20 Such type of equations were factorized in
Please study there. We will follow the same procedure. constant term = 20 has factors 1, 2, 4, 5, 10, 20 -1, -2, -4, -5, -10, -20. Let us verify whether f(a) = 0, where a = one of those factors. f(1) and f(-1) are not zero. f(2) = 23 - 5(2)2 - 4(2) + 20 = 8 - 20 - 8 + 20 = 0 ⇒ (x - 2) is a factor of f(x). ⇒ x = 2 is a root of f(x) = 0. one root is found as 2. Let the other roots be α and β s1 = 2 + α + β = -p1 = 5⇒ α + β = 3......(i)
s2 = (2)(α) + (2)(β) + (α)(β) = p2 = -4⇒ 2(α + β) + (α)(β) = -4.....(ii) Using (i) in (ii), we get 2(3) + (αβ) = -4 ⇒ αβ = -10.....(iii)
Using (i) and (iii), we can find (α - β) (α - β)2 = (α + β)2 - 4αβ= 32 - 4(-10) = 9 + 40 = 49 ⇒ (α - β) = ±7 ......(iv) (i) + (iv) gives 2α = 10 or -4 ⇒ α = 5 or -2. using these in (i), we get β = 3 - 5 or 3 + 2 = -2 or 5. These are same as values of α ⇒ the roots are 2, 5, -2. Let us verify whether these satisfy s3 or not. s3 = (2)(5)(-2) = -20 = -p3 [satisfied.]
Thus the roots of the given Algebra Equation are 2, 5, -2. Ans.
Example 2 of Polynomial Equation is thus solved.
Exercise : Polynomial Equation
Problems on Polynomial Equation :
Solve the Polynomial Equation 54x3 - 39x2 - 26x + 16 = 0, the roots of which are in G.P.
Solve the Algebra Equation x3 - 5x2 - 2x + 24 = 0
For Answers see at the bottom of the page.
Progressive Learning of Math : Polynomial Equation
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