Solve: x3 - 6x2+11x - 6 = 0 using the factor theorem.
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Step-by-Step Solution
Step 1: Identify the polynomial and possible factors
Let the given polynomial be P(x). According to the Factor Theorem, if (x-a) is a factor of P(x), then P(a) = 0. The possible integer roots are the divisors of the constant term, which is -6.
Step 2: Test for x=1
We test x=1 by substituting it into the polynomial P(x) .SinceP(1)=0, by the Factor Theorem, (x-1) is a factor of P(x)$.
Step 3: Perform polynomial division
Now that we know (x-1) is a factor, we can divide the polynomial P(x) by (x-1) to find the other factors. We perform polynomial long division or synthetic division.
Step 4: Factor the quadratic expression
The quadratic expression x2 - 5x + 6 can be factored into two linear factors. We look for two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3.
Step 5: State all factors and roots
Combining all the factors, we have P(x) = (x-1)(x-2)(x-3)$. Setting P(x) = 0 gives us the roots of the polynomial.