obtain all other zeroes of the polynomial x4 + 4x3 - 2x2 - 20x - 15
Answers
Solution:
Given:
→ f(x) = x⁴ + 4x³ - 2x² - 20x - 15
Put x = -1, we get:
→ f(-1) = (-1)⁴ + 4 × (-1)³ - 2 × (-1)² - 20 × (-1) - 15
= 1 - 4 - 2 + 20 - 15
= -5 + 20 - 15
= 20 - 20
= 0
Therefore, by factor theorem:
→ (x + 1) is a factor of f(x).
Divide f(x) by (x - 1):
x + 1) x⁴ + 4x³ - 2x² - 20x - 15 ( x³ + 3x² - 5x - 15
x⁴ + x³
–––––––––––––––––––––––––
3x³ - 2x²
3x³ + 3x²
–––––––––––––––––––––––––
-5x² - 20x
-5x² - 5x
–––––––––––––––––––––––––
-15x - 15
-15x - 15
–––––––––––––––––––––––––
0
Therefore:
→ f(x) = (x + 1)(x³ + 3x² - 5x - 15)
= (x + 1){x²(x + 3) - 5(x + 3)}
= (x + 1)(x + 3)(x² - 5)
Therefore:
→ (x + 1)(x + 3)(x² - 5) = 0
→ (x + 1) = 0 or (x + 3) = 0 or (x² - 5) = 0
When (x + 1) = 0:
→ x + 1 = 0
→ x = -1
When (x + 3) = 0:
→ x + 3 = 0
→ x = -3
When (x² - 5) = 0:
→ x² - 5 = 0
→ x² - (√5)² = 0
→ (x + √5)(x - √5) = 0
→ (x + √5) = 0 or (x - √5) = 0
→ x = √5, -√5
Therefore:
→ Zeros of f(x) are – (-1, -3, √5, -√5) (Answer)
Answer:
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