Factor each of the following polynomials.
1. x^2+x-6
2. 6x^2+13x-5
Thanx!
(x+3)(x-2)
3*5 = 15 which differs from 113 by 2 so
(3x-1)(2x+5)
To factor each of the polynomials, we need to find two binomials whose product is equal to the given polynomial. Here's how you can factor each of the given polynomials:
1. x^2 + x - 6:
To factor this polynomial, we need to find two numbers whose sum is equal to the coefficient of the middle term (which is 1) and whose product is equal to the product of the coefficient of the first term (which is 1) and the constant term (which is -6).
To do this, we can write all the possible pairs of numbers whose product is -6: (-1, 6), (1, -6), (-2, 3), (2, -3). Among these pairs, we want to find the pair whose sum is 1, which is (2, -3).
Now, we can rewrite the polynomial as the sum of these two pairs:
x^2 + 2x - 3x - 6
The first two terms, x^2 + 2x, can be factored as x(x + 2), and the last two terms, -3x - 6, can be factored as -3(x + 2):
x^2 + x - 6 = x(x + 2) - 3(x + 2)
Notice that we now have a common binomial factor, which is (x + 2). We can factor it out:
x^2 + x - 6 = (x - 3)(x + 2)
So, the factored form of the polynomial x^2 + x - 6 is (x - 3)(x + 2).
2. 6x^2 + 13x - 5:
To factor this polynomial, we follow the same process as before. We need to find two numbers whose sum is equal to the coefficient of the middle term (which is 13) and whose product is equal to the product of the coefficient of the first term (which is 6) and the constant term (which is -5).
To do this, we can write all the possible pairs of numbers whose product is -30 (the product of 6 and -5): (-1, 30), (1, -30), (-2, 15), (2, -15), (-3, 10), (3, -10), (-5, 6), (5, -6). Among these pairs, we want to find the pair whose sum is 13, which is (5, -6).
Now, we can rewrite the polynomial as the sum of these two pairs:
6x^2 + 5x - 6x - 5
The first two terms, 6x^2 + 5x, can be factored as x(6x + 5), and the last two terms, -6x - 5, can be factored as -1(6x + 5):
6x^2 + 13x - 5 = x(6x + 5) - 1(6x + 5)
Again, we have a common binomial factor, which is (6x + 5). Factoring it out, we get:
6x^2 + 13x - 5 = (x - 1)(6x + 5)
So, the factored form of the polynomial 6x^2 + 13x - 5 is (x - 1)(6x + 5).
By following these steps, you can factor any given polynomial by finding suitable pair(s) of numbers and factoring out common binomial factors.