I am having some difficulties factoring polynomials and I use only one method to figure the stuff out, which is the guess and check method because it's just mental math work and not that complicating.

Question

I am having some difficulties factoring polynomials and I use only one method to figure the stuff out, which is the guess and check method because it's just mental math work and not that complicating.

Here is a question that I know how to do, and below it I will list down the question that I do not know how to solve, and would like if you could solve it for me the way I solved the question above.

Factor:

2x(x+5)-3(x+5)
=2x^2+10x-3x-15
=2x^2+7x-15

Product=-15

(2x-5)(x+3)not this one

(2x+3)(x-5)not this one either

(2x-3)(x+5) this one is right

{So I basically used the guess and check method above after combining the like terms}

Now this is the question that left me puzzled. I would appreciate if you could solve it like the way I have done above with a simpler version.

9x^2+12xy+4y^2

{usually there is a constant in a polynomial, but this one doesn't have one, so I am cofused on what to do}

Answer 1

2x(x+5)-3(x+5)

Note that (x+5) is a common factor.
So
2x(x+5)-3(x+5)
=(x+5)(2x-3)

9x^2+12xy+4y^2
with the guess and check method, you can try to see if it is a perfect square:
√(9x²)=3x
√(4y²)=2y
So try
(3x+2y)²

Answer 2

That works out perfectly. Thank you very much MathMate.

Answer 3

To factor the polynomial 9x^2 + 12xy + 4y^2 using a similar method as you did before, we can look for two binomials that when multiplied together, give us the original polynomial.

First, let's identify the factors of 9x^2, which are 3x and 3x. Then, let's identify the factors of 4y^2, which are 2y and 2y. These will be the terms in our binomials.

Next, we need to determine the signs of the factors. We know that when multiplied, the middle term of the original polynomial, 12xy, is positive. This means that both factors need to have the same sign. Since the constant term is positive, the factors also need to be positive.

So far, we have:

(3x + ?)(3x + ?)

Now, we need to determine what goes in the question marks. To find these values, we need to consider the coefficient of the middle term, 12xy. It can be written as the sum of the products of the factors from the previous step:

12xy = (3x)(4y) + (3x)(2y) = 12xy + 6xy

This suggests that the question marks are 4y and 2y, respectively:

(3x + 4y)(3x + 2y)

Therefore, the factored form of the polynomial 9x^2 + 12xy + 4y^2 is (3x + 4y)(3x + 2y).