Factor the expression.
21x^2+55x+14
To factor the expression 21x^2 + 55x + 14, we need to find two binomials that, when multiplied together, equal the original expression.
We can start by looking at the coefficients of the expression, 21, 55, and 14, and determine if there is a common factor among them. In this case, the common factor is 7. By factoring out the greatest common factor, we get:
7(3x^2 + 11x + 2)
Next, we need to find two binomials that, when multiplied, equal the quadratic trinomial 3x^2 + 11x + 2. To find these binomials, we can use the factoring method or use the quadratic formula. In this case, let's use factoring.
We are looking for two binomials in the form (ax + b)(cx + d). To find the values of a, b, c, and d, we need to consider the coefficients of the quadratic trinomial. The first term, 3x^2, can be factored as (3x)(x). The last term, 2, can be factored as (1)(2) or (2)(1).
Now, we need to determine which combination of the factors will add up to the middle term, 11x. We see that (3x)(2) + (1)(x) = 6x + x = 7x. This tells us that the factors (3x + 1) and (x + 2) form our binomials.
Thus, the factored form of the expression 21x^2 + 55x + 14 is:
7(3x + 1)(x + 2)