I am not understanding how to factor polynomials. Example: Factor 12y^6-40y^5+28y^4

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12y^6-40y^5+28y^4

Take out the GCF..for example, each term has at LEASt a y^4 and the highest number it each can be divided by is 4 so..
4y^4(3y^2-10y+7)

To factor a polynomial like 12y^6-40y^5+28y^4, we can start by looking for common factors among the terms. In this case, notice that each term contains a common factor of 4y^4. We can factor it out using the distributive property:

4y^4(3y^2 - 10y + 7)

Now we have factored out the greatest common factor. Next, we can check if the remaining expression, 3y^2 - 10y + 7, can be factored further. Since it cannot be factored using simple integer coefficients, we can use the quadratic formula to find its factors. The quadratic formula states that for a quadratic equation in the form ax^2 + bx + c = 0, the roots (or factors) can be found using the formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Applying that to our equation:

y = (-(-10) ± √((-10)^2 - 4(3)(7))) / (2(3))

Simplifying:

y = (10 ± √(100 - 84)) / 6
y = (10 ± √16) / 6
y = (10 ± 4) / 6
y = (10 + 4) / 6 or y = (10 - 4) / 6
y = 14/6 or y = 6/6
y = 7/3 or y = 1

So the factors of 3y^2 - 10y + 7 are (y - 7/3) and (y - 1). Putting it all together, the factored form of the polynomial 12y^6-40y^5+28y^4 is:

4y^4(y - 7/3)(y - 1)