What are all the factors when the polynomial is completely factored?
12c^2-38c+20
To factor the quadratic polynomial completely, we look for two numbers that multiply to give the product of the leading coefficient (12) and the constant term (20) and that add to give the middle coefficient (-38).
Let's find these two numbers. The product of the leading coefficient and the constant term is 12 * 20 = 240. We need two numbers that multiply to 240 and add up to -38.
After trying different combinations, we find that -28 and -10 satisfy these conditions:
-28 * -10 = 240
-28 + -10 = -38
Now we can write the middle term of the polynomial -38c as -28c - 10c, splitting it into two terms:
12c^2 - 28c - 10c + 20
Next, we can factor by grouping. Group the terms to find the greatest common factor (GCF) for each group:
(12c^2 - 28c) - (10c - 20)
For the first group, the GCF is 4c. For the second group, the GCF is 10 (we take -10 to keep the sign correct in the grouping):
4c(3c - 7) - 10(3c - 7)
Now we can factor out the common binomial factor (3c - 7):
(4c - 10)(3c - 7)
So, the completely factored form of the polynomial 12c^2 - 38c + 20 is:
(4c - 10)(3c - 7)
You can check the factoring by expanding the product (4c - 10)(3c - 7) to ensure it results in the original polynomial.