I'm not sure how to factor 3x^3-12x^2+3x. I tried separatring the 3x so that 3x(x+1)(x-4, but I got -3x^3+-12x^2-3x^2+12x. Can you show me where I went wrong and how to properly factor this? Is there a general rule or tip for this sort of factoring with a number to the third power? Thanks!!
there is a common factor of 3x
so,
3x^3-12x^2+3x
= 3x(x^2 - 4x + 1)
The last part does not factor any more, so that's it.
To properly factor a polynomial like 3x^3 - 12x^2 + 3x, it's important to follow certain steps.
Step 1: Look for the greatest common factor (GCF) of all the terms. In this case, the GCF is 3x, so we can factor it out like this: 3x(x^2 - 4x + 1).
Step 2: Now you need to factor the quadratic expression x^2 - 4x + 1. This can be done using various methods such as factoring by grouping, completing the square, or using the quadratic formula. In this case, let's use the method of factoring by grouping:
First, multiply the coefficient of x^2 (which is 1) by the constant term (which is 1). The result is 1.
Next, we need to find two numbers that multiply to give 1 and add up to the coefficient of x (which is -4). Those numbers are -1 and -1.
So, we rewrite the middle term (-4x) as -1x - 1x:
x^2 - 1x - 1x + 1
Now we group the terms:
(x^2 - 1x) + (-1x + 1)
Factor out the GCF from each group:
x(x - 1) - 1(x - 1)
Notice that we have a common factor of (x - 1), which can be factored out:
(x - 1)(x - 1)
Step 3: Simplify the factors from step 1 and step 2:
3x(x - 1)(x - 1)
So, the properly factored form of the polynomial 3x^3 - 12x^2 + 3x is 3x(x - 1)(x - 1).
Regarding your question about a general rule or tip for factoring polynomials with a number to the third power, one approach is to first check for a common factor. Beyond that, factoring higher-degree polynomials often involves techniques like factoring by grouping, using the quadratic formula, or recognizing special factoring patterns (such as the difference of squares, perfect squares, and perfect cubes). Practice and familiarity with these techniques will help develop your ability to factor higher-degree polynomials effectively.