Factor x^3-8x^2+17x-10

Problem

x3−8*x2+17*x−10

Solution

1. Identify potential rational roots using the Rational Root Theorem. The possible integer roots are factors of the constant term −10 which are ±1,±2,±5,±10

2. Test the value x=1 by substituting it into the polynomial: 1−8*(1)2+17*(1)−10=1−8+17−10=0 Since the result is zero, (x−1) is a factor.

3. Divide the polynomial x3−8*x2+17*x−10 by (x−1) using synthetic division or long division to find the remaining quadratic factor.

4. Perform the division:

(x3−8*x2+17*x−10)÷(x−1)=x2−7*x+10

5. Factor the resulting quadratic expression x2−7*x+10 by finding two numbers that multiply to 10 and add to −7 These numbers are −2 and −5

6. Write the quadratic factor in its factored form:

x2−7*x+10=(x−2)*(x−5)

7. Combine all the factors to represent the original cubic polynomial.

Final Answer

x3−8*x2+17*x−10=(x−1)*(x−2)*(x−5)

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