Answer:
x³ - 3x² + 3x – 1 = (x – 1)(x – 1)(x – 1)
Step-by-step explanation:
Step 1. Use the rational root theorem to find a root.
The general formula for a third-degree polynomial is
f(x) = ax³ + bx² + cx + d
Your polynomial is
f(x) = x³ - 3x² + 3x - 1
a = 1; d = -1
Factors of a = ±1
Factors of d = ±1
Possible roots are x = 1 and x = -1.
f(1) = 1³ - 3(1²) + 3×1 -1
f(1) = 1 - 3×1 + 3×1 -1
f(1) = 1 - 3 + 3 -1
f(1) = 0
So, x - 1 is one root of the polynomial.
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Step 2. Use synthetic division to factor the polynomial.
1|1 -3 3 -1
| 1 -2 1
1 -2 1 0
So, f(x) = (x - 1)(x² - 2x +1)
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Step 3. Factor the quadratric
Find two numbers whose product is 1 and whose sum is -2.
The numbers are -1 and -1.
x² - 2x +1 = (x - 1)(x – 1)
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Step 4. Write out the factors.
x³ - 3x² + 3x – 1 = (x – 1)(x – 1)(x – 1)
