Factoring with two variables
Factor each polynomial
X^2-25x+150
Answer -
(x - 15)(x - 10)
Because -
15 * 10 = 150
-15 + -10 = -25
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To factor the quadratic polynomial x^2 - 25x + 150, we need to find two binomials that, when multiplied, give us the original expression.
First, we look for two numbers whose product is equal to the product of the coefficient of x^2 (which is 1) and the constant term (which is 150). In this case, the product is 1 * 150 = 150.
Next, we need to find two numbers whose sum is equal to the coefficient of x (which is -25). Let's find these numbers by listing all possible pairs of factors of 150:
1 and 150
2 and 75
3 and 50
5 and 30
6 and 25
10 and 15
Out of these pairs, we want the pair that adds up to -25. In this case, that would be 10 and 15.
Now we can rewrite the middle term of our original polynomial using these two numbers:
x^2 - 10x - 15x + 150
Now we can group the terms:
(x^2 - 10x) + (-15x + 150)
Next, we factor out the greatest common factor from each pair of terms:
x(x - 10) - 15(x - 10)
Now we have a common binomial factor of (x - 10), which we can factor out:
(x - 10)(x - 15)
Therefore, the factored form of the polynomial x^2 - 25x + 150 is (x - 10)(x - 15).