Which expression is the factored equivalent of 3x^2-4x+1?
A. (3x-1)(x-1)
B. (3x-1)(x+1)
C. (3x+1)(x-1)
D. (3x+1)(x+1)
When I factor it, I keep getting D. But I know the answer is A. So this must be a common mistake students make. What am I doing wrong?
Here is my work.
3x^2-4x+1
3x^2+3x+1x+1
(3x^2+3x)+(1x+1)
3x(x+1)+1(x+1)
(x+1)(3x+1)=(3x+1)(x+1)
Oh wait, never mind. I know exactly what I did wrong.
On the second line where you put 3x^2+3x+1x+1 it should really be 3x^2-3x-1x+1
You just lost track of your negative
You made a common mistake in your factoring process. Let's go through it step by step to help identify the error.
Starting with the expression: 3x^2 - 4x + 1.
1. First, determine the factors of the first term, 3x^2. The factors are 3x and x.
Rewriting the expression using these factors: (3x ) (x ).
2. Next, try to find two numbers that multiply to give the constant term, 1, and add up to the coefficient of the middle term, -4x.
In this case, the numbers that work are 1 and 1 (1 * 1 = 1 and 1 + 1 = 2).
Rewriting the expression using these numbers: (3x - 1) (x + 1).
So, the correct factored equivalent expression is (3x - 1) (x + 1), not (3x + 1) (x + 1).
Therefore, the correct answer is A. (3x - 1) (x - 1).
To factor the expression 3x^2-4x+1 correctly, let's follow the steps:
Step 1: Multiply the coefficient of the quadratic term (3) by the constant term (1) to get 3 * 1 = 3.
Step 2: Find two numbers that multiply to give 3 and add to give the coefficient of the linear term (-4). In this case, the numbers are -1 and -3 because (-1) * (-3) = 3 and (-1) + (-3) = -4.
Step 3: Rewrite the middle term -4x using the two numbers found in step 2.
3x^2 - x - 3x + 1
Step 4: Group the terms and factor by grouping.
(3x^2 - x) + (-3x + 1)
Step 5: Take out the greatest common factor from each group.
x(3x - 1) - 1(3x - 1)
Step 6: Notice that (3x - 1) appears in both terms. Factor it out.
(3x - 1)(x - 1)
Therefore, the factored equivalent of 3x^2-4x+1 is (3x - 1)(x - 1), which corresponds to option A. It is essential to be careful and accurate while grouping and factoring to avoid common mistakes.