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# is there an easier way to do this without multiplying straight through?

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Dec 8, 2018

#1
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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

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

x = -3 or x = 4 + i or x - 4 = -i

x = -3          or x = 4 + i          or x = 4 - i

Dec 8, 2018
#2
+2

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   Dec 8, 2018