In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:
f(x) = x⁵+3x⁴-x³-3x²+5x+15, g(x) = x+3
Answers
Given : f(x) = x⁵ + 3x⁴ - x³ - 3x² + 5x + 15, g(x) = x + 3
To find : Whether polynomial g(x) is a factor of polynomial f(x) or, not.
In order to find whether g (x) = x – (-3) is a factor of polynomial f (x) or not, it is sufficient to prove that f (- 3) = 0
Now,
f(x) = x⁵ + 3x⁴ - x³ - 3x² + 5x + 15
f (- 3) = (- 3)⁵ + 3 (- 3)⁴ – (- 3)³ – 3 (- 3)² + 5 (- 3) + 15
f (- 3) = - 243 + 243 – (- 27) – 3 (9) + 5 (- 3) + 15
f (- 3) = - 243 + 243 + 27 – 27 – 15 + 15
f (- 3) = 0
Hence, g (x) is a factor of f (x).
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Answer:
For the polynomial f(x) to be a factor of the other polynomial g(x), the remainder of when the former is divided by the latter should be equal to zero.
g(x) = x+3 = 0
=> x = -3
f(x) = x⁵+3x⁴-x³-3x²+5x+15
=> f(-3) = (-3)⁵+3(-3)⁴-(-3)³-3(-3)²+5(-3)+15
=> f(-3) = -243 + 243 + 27 - 27 - 15 + 15
=> f(-3) = 0
Thus, this is a factor of f(x).
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