Factor.
14x^4 - 19x^3 - 3x^2
I am having a little trouble with this one. Please help if you can!
Thank you! :-)
To factor the given expression: 14x^4 - 19x^3 - 3x^2, we need to look for common factors and then use factoring techniques.
First, look for the greatest common factor (GCF) of the coefficients. In this case, the GCF of 14, -19, and -3 is 1.
Next, look for the GCF of the variables (x). In this case, the GCF of x^4, x^3, and x^2 is x^2.
Now, factor out the GCF from each term:
14x^4 - 19x^3 - 3x^2 = x^2(14x^2 - 19x - 3)
The expression inside the parentheses is a quadratic trinomial. To factor it further, we can look for two binomials whose product equals the quadratic trinomial.
In general, a quadratic trinomial has the form ax^2 + bx + c. We need to find two numbers, let's call them m and n, such that the following conditions are met:
1. Their product is equal to the product of a and c: m * n = ac
2. Their sum is equal to b: m + n = b
In this case, a = 14, b = -19, and c = -3. We need to find two numbers whose product is equal to ac (14 * -3 = -42) and whose sum is equal to b (-19).
After trying different factor pairs of -42, we find that -21 and 2 satisfy both conditions:
-21 * 2 = -42
-21 + 2 = -19
Now, rewrite the quadratic trinomial using -21x and 2x as middle terms:
14x^2 - 19x - 3 = 14x^2 - 21x + 2x - 3
Next, group the terms:
(14x^2 - 21x) + (2x - 3)
Now, factor out the greatest common factor from each group:
7x(2x - 3) + 1(2x - 3)
Notice that (2x - 3) appears as a common factor. We can factor it out:
(7x + 1)(2x - 3)
Therefore, we have factored the expression 14x^4 - 19x^3 - 3x^2 as (7x + 1)(2x - 3).