Factoring a quadratic polynomial in two variables
3x^2+17xy+20y^2
see other post. It is not necessary nor desirable to post a question more than once.
To factor a quadratic polynomial in two variables like 3x^2 + 17xy + 20y^2, we will look for two binomials that when multiplied together will give us the original polynomial.
Step 1: Check for any common factors
In this case, there are no common factors among the coefficients of the polynomial.
Step 2: Identify the pattern
Look at the coefficients of the quadratic terms (3x^2) and (20y^2). Try to find two numbers that multiply to give the product of these coefficients (3 * 20 = 60) and also add up to the coefficient of the linear term (17xy).
Step 3: Apply the pattern
The numbers that multiply to give 60 and add up to 17 are 12 and 5. We can rewrite the middle term 17xy as 12xy + 5xy.
Now, we rewrite our polynomial using these new terms:
3x^2 + 17xy + 20y^2
= 3x^2 + 12xy + 5xy + 20y^2
Step 4: Group and factor
Now, we group the terms and factor them separately:
(3x^2 + 12xy) + (5xy + 20y^2)
= 3x(x + 4y) + 5y(x + 4y)
Step 5: Combine the factors
Since both terms now have a common factor of (x + 4y), we can combine them into a single expression:
(3x + 5y)(x + 4y)
Therefore, the factored form of the quadratic polynomial 3x^2 + 17xy + 20y^2 is (3x + 5y)(x + 4y).