Solve for x:
x^3 - 5 x^2 - 7 x + 51 = 0
The left hand side factors into a product with two terms:
(x + 3) (x^2 - 8 x + 17) = 0
Split into two equations:
x + 3 = 0 or x^2 - 8 x + 17 = 0
Subtract 3 from both sides:
x = -3 or x^2 - 8 x + 17 = 0
Subtract 17 from both sides:
x = -3 or x^2 - 8 x = -17
Add 16 to both sides:
x = -3 or x^2 - 8 x + 16 = -1
Write the left hand side as a square:
x = -3 or (x - 4)^2 = -1
Take the square root of both sides:
x = -3 or x - 4 = i or x - 4 = -i
Add 4 to both sides:
x = -3 or x = 4 + i or x - 4 = -i
Add 4 to both sides:
x = -3 or x = 4 + i or x = 4 - i
+128474 If 4 + i a root then so is the conjugate 4 - i
Unfortunately.....We have to multiply these....!!!!
Multiplying these we get the polynomial
(x - (4 + i) ) ( x - (4 - i) ) = x^2 - x(4 + i) - x(4 - i) + (4 + i)(4-i) =
x^2 - 8x + 16 - i^2 =
x^2 - 8x + 17
To find the remaining polynomial....we can perform some polynomial division
x + 3
x^2 - 8x + 17 [ x^3 - 5x^2 - 7x + 51 ]
x^3 - 8x^2 + 17x
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3x^2 - 24x + 51
3x^2 - 24x + 51
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Since x + 3 is the remaining polynomial, x = -3 is the remaining root
