factor (2x + 1)
To factor 2x + 1, we need to find two binomials whose product is equal to 2x + 1.
Let's try to express 2x + 1 as the product of two binomials:
(2x + 1) = (ax + b)(cx + d)
Using the distributive property, we can determine that:
(2x + 1) = acx^2 + (ad + bc)x + bd
For 2x + 1 to be factored, the coefficient of x^2, ac, must be equal to 0, and the coefficient of x, ad + bc, must be equal to 2, and the constant term, bd, must be equal to 1.
Since ac must be 0, either a or c must be zero. Let's assign a = 2 and c = 1:
(2x + 1) = (2x + b)(x + d)
Now, we just need to find the appropriate values for b and d.
To find b and d, we multiply the two binomials and compare it with the original expression:
(2x + 1) = (2x + b)(x + d)
(2x + 1) = 2x(x + d) + b(x + d)
(2x + 1) = 2x^2 + 2dx + bx + bd
(2x + 1) = 2x^2 + (2d + b)x + bd
From this comparison, we can see that:
2x^2 + (2d + b)x + bd = 2x + 1
Comparing the coefficients, we can set up the following equations:
2d + b = 2 --> Equation 1
bd = 1 --> Equation 2
From Equation 2, we know that b and d must be factors of 1. The factors of 1 are 1 and -1. Let's start with b = 1 and d = 1.
Plugging in these values to Equation 1, we have:
2(1) + 1 = 2 --> The equation is true.
Therefore, b = 1 and d = 1.
Substituting these values back into our assumed binomial factorization, we get:
(2x + 1) = (2x + 1)(x + 1)
Hence, the factorization of 2x + 1 is (2x + 1)(x + 1).