Factoring a quadratic polynomial in two variables
3x^2+17xy+20y^2
I know if I multiply 5x4 I get 20, and that 3x4 + 5 is 17.
(x + 4y)(_x + 5y)
Fill the blank with the only number that works.
Note the _x^2 must equal 3x^2, and (as BobPurley has stated) 5 + 4*3 = 17
To factor a quadratic polynomial in two variables, such as 3x^2+17xy+20y^2, we need to find two binomial factors that, when multiplied, give us the original polynomial.
Here's how we can do it step by step:
Step 1: Look at the coefficient of x^2, which is 3 in this case.
Step 2: Determine the factors of 3. In this case, it is 1 and 3.
Step 3: Look at the constant term, which is 20 in this case.
Step 4: Determine the factors of 20. In this case, it can be 1 and 20, 2 and 10, or 4 and 5.
Now, we need to figure out which pair of factors will give us the middle term, which is 17xy. We need to find two numbers whose product is the product of the coefficient of xy (17) and the constant term (20).
Let's try the pair of factors (4, 5):
The sum of 4 and 5 is 9, and the difference is 1. Since the middle term is positive (17xy), we want the larger number (5) to have the positive sign. Therefore, we can rewrite the middle term as 5xy + 12xy.
So, the factored form of the quadratic polynomial 3x^2+17xy+20y^2 using the pair of factors (4, 5) is:
(3x + 4y)(x + 5y)
Thus, 3x^2+17xy+20y^2 factors into (3x + 4y)(x + 5y).