Consider the sum of rational expressions: 11−2z/z^2−6z+5 + z^2−3/z−5
.
The trinomial z^2−6z+5
can be factored into two binomials. Write the trinomial in factored form
The trinomial z^2−6z+5 can be factored into (z-5)(z-1) in factored form.
To factor the trinomial z^2 - 6z + 5, we are looking for two binomials in the form (z - a)(z - b) whose product gives us the original trinomial.
The product of two binomials is given by the following expression:
(z - a)(z - b) = z^2 - (a + b)z + ab.
We need to find two values, a and b, such that their sum is -6 and their product is 5. Let's list some pairs of integers whose product is 5:
1 * 5 = 5
-1 * -5 = 5
The only pair that will give us a sum of -6 is -1 and -5. Therefore, we can factor the trinomial as:
z^2 - 6z + 5 = (z - 1)(z - 5).
So, the factored form of the trinomial z^2 - 6z + 5 is (z - 1)(z - 5).
To factor the trinomial z^2 - 6z + 5, we need to find two binomials whose product is equal to the trinomial. Here's how you can do it:
Step 1: Identify the factors of the first term, z^2. In this case, z and z are the only factors.
Step 2: Identify the factors of the last term, 5. The factors of 5 are 1 and 5, or -1 and -5.
Step 3: Find the combination of factors that, when multiplied, give the middle term, -6z.
To do this, calculate the sums of each pair of possible factors: z + 1, z + 5, z - 1, and z - 5.
In this case, the combination that adds up to -6z is z - 1 and z - 5 because (z - 5) + (z - 1) = 2z - 6z = -6z.
So, the factored form of the trinomial z^2 - 6z + 5 is (z - 1)(z - 5).