I need help with factoring polynomials using pi and sigma. I understand it, but what is the best way to find the sigma when the terms are, in my opinion, really large? It takes me forever to find it, guessing.
Let's say the problem is this:
56x² - 53xy + 12y²
The pi = 672 = (-____)(-_____)
and the sigma = -53 = (-__) + (-___)
How would you solve the problem?
If the method is not convenient then you should not use.
The factorization is of the form
A (x + p1 y) (x + p2 y)
with A = 56.
If you equate the polynomial to zero and solve for x, the solutions are:
x = -p1 y and x = -p2 y
So, we substitute x = -p y in the polynomial and equate it to zero:
(56 p^2 + 53 p + 12) y^2 = 0
We know that if p = p1 or p2, this will be zero for nonzero y.
So, p1 and p2 are the solutions of the equation:
56 p^2 + 53 p + 12 = 0
p = (-53 +-11)/(2*56)
To factor the given polynomial, 56x² - 53xy + 12y², using the pi and sigma method, you need to find two numbers whose product (the pi) is equal to the product of the leading coefficient (in this case, 56) and the constant term (in this case, 12), and whose sum (the sigma) is equal to the coefficient of the middle term (in this case, -53).
Let's start by finding the pi:
The pi is equal to the product of the leading coefficient (56) and the constant term (12), so pi = 56 * 12 = 672.
Now, we need to find two numbers whose product is 672. Since 672 is a relatively large number, it can be challenging to find the exact factors by guessing. In this case, you can use a method called prime factorization to help you identify the factors.
Prime factorization involves breaking down a number into its prime factors. To do this, divide the number by its smallest prime factor repeatedly until you are left with all prime factors.
Let's prime factorize 672:
Start by dividing 672 by the smallest prime number, which is 2:
672 ÷ 2 = 336
Now, keep dividing the result (336) by 2 until you can no longer divide evenly:
336 ÷ 2 = 168
168 ÷ 2 = 84
84 ÷ 2 = 42
42 ÷ 2 = 21
Now, you're left with 21, which is an odd number. The next smallest prime number is 3:
21 ÷ 3 = 7
So, the prime factorization of 672 is 2^5 * 3 * 7.
Now, you can use these prime factors to find the factors of 672 that add up to -53. Take note that the exponents of the prime factors must add up to the number of variables in the polynomial, which is 2 in this case.
Since the sigma is negative (-53), one of the factors must be negative. To find the two factors whose sum is -53, you can try different combinations of the prime factors.
Possible combinations in this case are:
(-1) * (672)
(-2) * (336)
(-3) * (224)
(-4) * (168)
(-6) * (112)
(-7) * (96)
(-8) * (84)
(-12) * (56)
(-14) * (48)
(-16) * (42)
(-21) * (32)
(-24) * (28)
Among these combinations, you need to find the pair whose sum is -53.
In this case, the pair (-8) and (84) will give a sum of -8 + 84 = 76, which is different from -53. So, this pair is not the correct pair to use in factoring.
You need to continue trying different combinations until you find the correct pair. In this example, the correct pair is (-3) and (224) because -3 + 224 = 221, and by rearranging the signs, you can get -224 + 3 = -221, which is close to -53.
So the factored form of the polynomial 56x² - 53xy + 12y² is:
56x² - 53xy + 12y² = (8x - 3y)(7x - 4y).
Remember, the prime factorization method may not always be the fastest method for finding the factors of large numbers. However, it can be a helpful approach when you are faced with large numbers and need to factor the polynomial using the pi and sigma method.