High School

Find the zeros of the following polynomial function.

If [tex]f(x) = x^{4} + x^{3} + 23x^{2} + 25x - 50[/tex]

Answer :

The roots of the polynomial f(x) = x⁴ + x³ + 23x² + 25x - 50 are: 1, -2, +5i and -5i

How to find the roots of the polynomial function?

The polynomial function is given as:

f(x) = x⁴ + x³ + 23x² + 25x - 50

Let us try x = 1:

f(1) = (1)⁴ + (1)³ + 23(1)² + 25(1) - 50

f(1) = 1 + 1 + 23 + 25 - 50

f(1) = 0

Thus, 1 is a root and the factor is (x - 1)

Now, let us try x = -2:

f(-2) = (-2)⁴ + (-2)³ + 23(-2)² + 25(-2) - 50

f(-2) = 16 - 8 + 92 - 50 - 50

f(-2) = 0

Thus, -2 is a root and the factor is (x + 2)

Multiplying both factors gives us:

(x - 1)(x + 2) = x² + x - 2

Thus, let us use long division to simplify the polynomial:

x² + 25

x² + x - 2 | x⁴ + x³ + 23x² + 25x - 50

- x⁴ + x³ - 2x²

25x² + 25x - 50

- 25x² + 25x - 50

0

Simplifying to find the remaining roots gives:

x² + 25 = 0

x² = -25

x = ±√-25

x = ±5i

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