I am trying to find all solutions to this algebra (factoring) problem, x3 – 3x2 – x + 3 = 0, and I keep getting the wrong answer. Please help!
This equation is a great candidate for factoring by grouping. Why? Factoring by grouping is a method usually done on polynomials with four or more terms — usually with an even number. Also, factoring by grouping works well when there is no common factor for all the terms in the polynomial, but there are common factors in pairs of the terms.
To factor by grouping, the first step is to rewrite the polynomial in groups:
x3 – 3x2 – x + 3 = 0 (x3 – 3x2) – (x – 3) = 0
There is a common factor of x2 in the first pair, so factor it out:
x2(x – 3) – (x – 3) = 0
You can see that each pair has a common factor of (x – 3). After you group, if you don't have a common factor in each pair, try rearranging the terms in another way. If you still don't end up with a common factor in each pair, it may be that the equation can't factored (or you've made a mistake — be sure to double check your work!)
Since there is a common factor, factor (x – 3) out of the two groups:
(x – 3)(x2 – 1) = 0
Now set each binomial equal to 0 and solve:
x – 3 = 0 x2 – 1 = 0 x = 3 (x – 1)(x + 1) = 0 x = 3 OR x = 1 OR x = –1
Check these three possible solutions by substituting the values for x back into the original equation. You should find that all three solutions are valid!
