Directions: Facotor each of the following polynomials completely.
14x ^2 -20x + 6
the answer that i got was:
7x^2-10x+3= (7x-3)(x-1)
not quite. You left out the two, you just cant throw it away.
2(7x-3)(x-1)
can you show me please the steps I am getting confused.
Sure! I'll be happy to explain the steps to help you factor the polynomial correctly.
To begin factoring the polynomial 14x^2 - 20x + 6 completely, we'll use a method called factoring by grouping. Here's how you can do it:
Step 1: Look for the greatest common factor (GCF) of all the terms. In this case, all the terms have a GCF of 2. So, we can factor it out first:
2(7x^2 - 10x + 3)
Step 2: Now, we need to factor the remaining quadratic trinomial inside the parentheses, 7x^2 - 10x + 3. To do this, we'll find two numbers whose product is the product of the coefficient of x^2 term (7x^2) and the constant term (3). In this case, the product is 7 * 3 = 21.
Step 3: Next, we need to find two numbers whose sum is equal to the coefficient of the x term (-10x). We need to find two numbers that, when multiplied, give us 21 and, when added, give us -10. In this case, the numbers are -3 and -7 because (-3) * (-7) = 21 and (-3) + (-7) = -10.
Step 4: Now, we use these numbers to split the middle term (-10x) into two terms (-3x and -7x).
So, the expression becomes:
2(7x^2 - 3x - 7x + 3)
Step 5: Now, we'll factor by grouping. We group the first two terms together and the last two terms together:
2((7x^2 - 3x) + (-7x + 3))
Step 6: Next, we factor out the GCF from each group:
2( x(7x - 3) - 1(7x - 3))
Step 7: Now, notice that we have a common binomial factor of (7x - 3) in both terms. We can factor it out:
2( (7x - 3)(x - 1) )
So, the completely factored form of the polynomial 14x^2 - 20x + 6 is 2(7x - 3)(x - 1).
I hope this explanation helps! Let me know if you have any further questions.