How would you find all the solutions of x^5 - 3x^4 - 2x^3 - 4x^2 - 24x + 32
+23252 To find rational answers, factor the constant term (32) and divide each of these numbers by the factors of the coefficient of the iinitial term (1).
The factors of 32 are +1, -1, +2, -2, +4, -4, +8, -8. +16. -16. +32, and -32.
Try these, one at a time, until you find one that makes the expresssion zero.
If you try +1, the expression becomes zero; this tells you that one of the answers is +1, and one of the factors of the expression is (x - 1).
Now, there are a couple of things that you can do.
You can divide the original expression by (x - 1) to give you a smaller expression:
x4 - 2x3 -4x2 - 8x - 32
Again, factor the constant term (32) and divide each of these numbers by the facotrs of the coefficient of the initial term (1).
And again, these factor are +1, -1, +2, -2, +4, -4, +8, -8. +16. -16. +32, and -32.
Try these, one at a time, until you find one that makes the expression zero.
This will give you a second solution ... create the factor .... divide, and continue this process until you can no longer get another solution.
Solve for x:
x^5 - 3 x^4 - 2 x^3 - 4 x^2 - 24 x + 32 = 0
The left hand side factors into a product with four terms:
(x - 4) (x - 1) (x + 2) (x^2 + 4) = 0
Split into four equations:
x - 4 = 0 or x - 1 = 0 or x + 2 = 0 or x^2 + 4 = 0
Add 4 to both sides:
x = 4 or x - 1 = 0 or x + 2 = 0 or x^2 + 4 = 0
Add 1 to both sides:
x = 4 or x = 1 or x + 2 = 0 or x^2 + 4 = 0
Subtract 2 from both sides:
x = 4 or x = 1 or x = -2 or x^2 + 4 = 0
Subtract 4 from both sides:
x = 4 or x = 1 or x = -2 or x^2 = -4
Take the square root of both sides:
Answer: |x = 4 or x = 1 or x = -2 or x = 2 i or x = -2 i
+14995 How would you find all the solutions of x^5 - 3x^4 - 2x^3 - 4x^2 - 24x + 32
\( x^5 - 3x^4 - 2x^3 - 4x^2 - 24x + 32 \) is a term. There is no solution.
I guess you mean a function and you're looking for the zeros.
\(f(x)= x^5 - 3x^4 - 2x^3 - 4x^2 - 24x + 32 =0\)
Solution guess and simplify term several times by polynomial division.
\(x_1=-2\)
\(x_2=1\)
\(x_3=4\)
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