obtain all other zeros of the polynomial x to the power 4 + 4 x to the power 3 - 2 X ^ 2 - 20x- 15 if two of its zeros are root 5 and minus root 5
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Answers
Your answer is on the given attachment
Answer:
√5, -√5, -1, -3
Step-by-step explanation:
Let f(x) = x⁴ + 4x³ - 2x² - 20x - 15.
Given, √5 and -√5 are the zeroes of the polynomial.
Hence, (x + √5)(x - √5) are the zeroes of the polynomial.
⇒ x² - 5 is a factor of the polynomial.
Now, we divide the polynomial by x² - 5.
Long Division Method:
x² - 5) x⁴ + 4x³ - 2x² - 20x - 15 (x² + 4x + 3
x⁴ - 5x²
----------------------------------
4x³ + 3x² - 20x
4x³ - 20x
------------------------------------
3x² - 15
3x² - 15
-------------------------------------
0
Now,
⇒ x² + 4x + 3 = 0
⇒ x² + x + 3x + 3 = 0
⇒ x(x + 1) + 3(x + 1) = 0
⇒ (x + 1)(x + 3) = 0
⇒ x = -1, -3.
Therefore, the zeroes of the polynomial are √5, -√5, -1,-3
Hope it helps!