One of the polynomials below cannot be factored using integers. Write “prime” for the one trinomial that does not factor and all the factors for the two that do.
a) x2– 5x + 6
b) x2+ x + 6
c) 2x2– 2x + 12
a) which two factors of 6 add to 5?
b) which two factors of 6 differ by 1?
c) which two factors of 6 add to 1?
To determine if a polynomial can be factored using integers, we need to check if it is prime or if it can be expressed as a product of two or more linear factors with integer coefficients.
a) x^2 - 5x + 6
To factor this trinomial, we need to find two numbers that multiply to give 6 and add up to -5. The numbers that satisfy this are -2 and -3, so the factored form of the trinomial is (x - 2)(x - 3).
b) x^2 + x + 6
We need to find two numbers that multiply to give 6 and add up to 1. Unfortunately, there are no such integers. Therefore, this trinomial cannot be factored using integers. It is a prime polynomial.
c) 2x^2 - 2x + 12
Let's divide all the coefficients by their greatest common factor, which is 2. We get:
x^2 - x + 6
Now, we need to find two numbers that multiply to give 6 and add up to -1. The numbers that satisfy this are -2 and -3, so the factored form of the trinomial is (x - 2)(x - 3).
In summary:
a) The factors for x^2 - 5x + 6 are (x - 2)(x - 3).
b) The polynomial x^2 + x + 6 is a prime polynomial and cannot be factored using integers.
c) The factors for 2x^2 - 2x + 12 are 2(x - 2)(x - 3).