factorize
x^3-3x^2+3x+7
Look at the coefficients as
1 -3 3 7
They add up to 8, so x=1 is NOT a factor.
Now change the sign of the coefficients of the odd powers (x^3 and x):
-1 -3 -3 7
They add up to zero, so x=-1 is a factor.
We do a synthetic division, or a long division to get:
(x^3-3x^2+3x+7)/(x+1)=x^2-4x+7
which has no rational or real factors.
So the factorization is complete.
(x^3-3x^2+3x+7)=(x+1)(x^2-4x+7)
To factorize the expression x^3 - 3x^2 + 3x + 7, we can use the method of synthetic division or long division to find a rational root, and then apply synthetic division. However, in this case, it seems that there is no rational root.
Alternatively, we can make use of grouping to factorize the expression. Let's proceed with this method:
Step 1: Group the terms
(x^3 - 3x^2) + (3x + 7)
Step 2: Factor out the Greatest Common Factor (GCF) from each group
x^2(x - 3) + 1(3x + 7)
Step 3: Combine the terms back together
x^2(x - 3) + 1(3x + 7)
Step 4: We have now successfully factored the expression as:
(x^2 + 1)(x - 3) + 7
Therefore, the factorization of x^3 - 3x^2 + 3x + 7 is (x^2 + 1)(x - 3) + 7.