To determine the factored form of the polynomial [tex]\( z^2 - 10z + 25 \)[/tex], we will follow a series of logical steps.
1. Identify the polynomial: [tex]\( z^2 - 10z + 25 \)[/tex].
2. Recognize that this polynomial is a quadratic expression in the form [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -10\)[/tex]
- [tex]\(c = 25\)[/tex]
3. Factor the polynomial by finding two binomials that multiply together to give the original polynomial. These binomials should take the form [tex]\((z - p)(z - q)\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are numbers that satisfy the following conditions:
- [tex]\( p \cdot q = c = 25 \)[/tex]
- [tex]\( p + q = b = -10 \)[/tex]
4. Find the values of p and q:
- We need two numbers that multiply to 25 and add up to -10.
- These numbers are: -5 and -5 because:
- [tex]\((-5) \cdot (-5) = 25\)[/tex]
- [tex]\((-5) + (-5) = -10\)[/tex]
5. Write the factored form:
- Hence, the polynomial can be factored as [tex]\( (z - 5)(z - 5) \)[/tex].
- This can also be written as [tex]\( (z - 5)^2 \)[/tex].
So, the factored form of [tex]\( z^2 - 10z + 25 \)[/tex] is:
[tex]\[
(z - 5)(z - 5)
\][/tex]
or equivalently:
[tex]\[
(z - 5)^2
\][/tex]
