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in Polynomials by (56.3k points)

Using factor theorem, factorize the polynomials:

x3 – 6x2 + 3x + 10

1 Answer

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Best answer

Let f(x) = x3 – 6x2 + 3x + 10

Constant term = 10

Factors of 10 are ±1, ±2, ±5, ±10 

Let x + 1 = 0 or x = -1 

f(-1) = (-1)3 – 6(-1)2 + 3(-1) + 10 

= 10 – 10 

= 0 

f(-1) = 0 

Let x + 2 = 0 or x = -2 

f(-2) = (-2)3 – 6(-2)2 + 3(-2) + 10 

= -8 – 24 – 6 + 10 

= -28 

f(-2) ≠ 0 

Let x – 2 = 0 or x = 2 

f(2) = (2)3 – 6(2)2 + 3(2) + 10 

= 8 – 24 + 6 + 10 

= 0 

f(2) = 0 

Let x – 5 = 0 or x = 5 

f(5) = (5)3 – 6(5)2 + 3(5) + 10

= 125 – 150 + 15 + 10 

= 0 

f(5) = 0 

Therefore, (x + 1), (x – 2) and (x - 5) are factors of f(x) 

Hence f(x) = (x + 1) (x – 2) (x - 5)

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