If the Algebra Equation x4 + 4x3 - 2x2 - 12x + 9 = 0, has a pair of equal roots, solve the equation.
Solution to Example 1 on Algebra Puzzles : The given Algebra Equation is x4 + 4x3 - 2x2 - 12x + 9 = 0 Comparing this with x4 + p1x3 + p2x2 + p3x + p4 = 0, we get p1 = 4; p2 = -2; p3 = -12; p4 = 9
Also we have s1 = -p1 = -4;s2 = p2 = -2;s3 = -p3 = 12;s4 = p4 = 9;
By data two pairs of roots are equal. ∴ Let the roots be α, α, β, β s1 = α + α + β + β = 2(α + β) = -4⇒ α + β = -2........(i)
s4 = (α)(α)(β)(β) = (αβ)2 = 9⇒ αβ = ±3 ......(ii)
Using (i) and (ii), we get (α - β)2 = (α + β)2 - 4αβ= (-2)2 - 4(±3) = 4 - 12 or 4 + 12 = -8 or 16 Taking positive value, α - β = ±4.......(iii) (i) + (iii) gives 2α = 2 or -6⇒ α = 1 or -3 using these in (i), we get β = -2 - α = -2 -1 or -2 +3 = -3 or 1 These are the same as α values. ∴ the roots are 1, 1, -3, -3.
These roots are found using s1 and s4. Let us verify whether these satisfy s2 and s3. s2 = (1)(1) + (1)(-3) + (1)(-3) + (1)(-3) + (1)(-3) + (-3)(-3) = 10 - 12 = -2 [satisfied.] s3 = (1)(1)(-3) + (1)(1)(-3) + (1)(-3)(-3) + (1)(-3)(-3) = 18 - 6 = 12 [satisfied.]
Thus the roots of the given Algebra Equation are 1, 1, -3, -3. Ans.
Thus Example 1 on Algebra Puzzles is solved.
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Solve the Algebra Equation 16x4 - 64x3 + 56x2 + 16x - 15 = 0, the roots of which are in A.P.
Solution to Example 2 on Algebra Puzzles : The given Algebra Equation is 16x4 - 64x3 + 56x2 + 16x - 15 = 0 Dividing both sides of the equation by 16, we get x4 - 4x3 + (7⁄2)x2 + x - 15⁄16 = 0 Comparing this with x4 + p1x3 + p2x2 + p3x + p4 = 0, we get p1 = -4; p2 = 7⁄2; p3 = 1; p4 = -15⁄16
By data the roots are in A.P. Let the roots be α - 3β, α - β, α + β, α + 3β We know, s1 = -p1 = 4⇒ α - 3β + α - β + α + β + α + 3β = 4⇒ 4α = 4⇒ α = 1......(i)
Using the value of α = 1, the roots become 1 - 3β, 1 - β, 1 + β, 1 + 3β
When β = +(1⁄2), the roots become -1⁄2, 1⁄2, 3⁄2, 5⁄2. When β = -(1⁄2), the roots become 5⁄2, 3⁄2, 1⁄2, -1⁄2. As we can see, both sets are same.
Let us verify whether the set of roots satisfy values of s3 and s4We know, s3 = -p3 = -1 and s4 = p4 = -15⁄16But, s3 = (-1⁄2)(1⁄2)(3⁄2) + (-1⁄2)(1⁄2)(5⁄2) + (-1⁄2)(3⁄2)(5⁄2) + (1⁄2)(3⁄2)(5⁄2) = (-3⁄8) + (-5⁄8) + (-15⁄8) + (15⁄8) = -1 [satisfied.]s4 = (-1⁄2)(1⁄2)(3⁄2)(5⁄2) = -15⁄16 [satisfied.]
∴ The roots of the given Algebra Equation are -1⁄2, 1⁄2, 3⁄2, 5⁄2. Ans.
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