The trinomial z2−6z+5
can be factored into two binomials. Write the trinomial in factored form.
The factored form of the trinomial is (z-1)(z-5)
To factor the trinomial z^2 - 6z + 5, we need to find two binomials whose product equals the given trinomial.
First, we need to find two numbers that multiply to give 5 (the constant term) and add up to -6 (the coefficient of the linear term).
The factors of 5 are 1 and 5.
The two numbers that add up to -6 are -1 and -5.
Therefore, the factored form of the trinomial is (z - 1)(z - 5).
To factor the given trinomial, we are looking for two binomials in the form (z + a)(z + b) that multiply together to give the original trinomial.
In this case, we have the trinomial: z^2 - 6z + 5.
To find the factors, we need to identify two numbers, let's call them a and b, such that their sum is equal to the coefficient of the middle term (-6) and their product is equal to the constant term (5).
In this case, the possible pairs of numbers that satisfy these conditions are:
(1, 5) and (-1, -5)
Now let's try each pair and check if their product matches the constant term and their sum matches the coefficient of the middle term:
For the pair (1, 5):
(1)(5) = 5 (product matches)
1 + 5 = 6 (sum matches)
For the pair (-1, -5):
(-1)(-5) = 5 (product matches)
-1 + (-5) = -6 (sum matches)
Since both pairs of numbers satisfy the conditions, we can choose either pair to factor the trinomial.
Let's choose the pair (1, 5) and rewrite the middle term (-6z) using these numbers:
z^2 - z + 5z + 5
Now we will group the terms and factor out the greatest common factor from each group:
(z^2 - z) + (5z + 5)
z(z - 1) + 5(z + 1)
Finally, we have the factored form of the trinomial:
(z - 1)(z + 5)