Explore IDNLearn.com's extensive Q&A database and find the answers you need. Our platform is designed to provide trustworthy and thorough answers to any questions you may have.
Sagot :
To factor the polynomial [tex]\( z^2 - 10z + 25 \)[/tex]:
1. Identify the polynomial structure: It is a quadratic polynomial of the form [tex]\( az^2 + bz + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 25 \)[/tex].
2. Check if it can be factored by recognizing perfect squares:
- A perfect square trinomial takes the form [tex]\( (z - p)^2 \)[/tex], which expands to [tex]\( z^2 - 2pz + p^2 \)[/tex].
- Here, compare the given polynomial [tex]\( z^2 - 10z + 25 \)[/tex] to [tex]\( z^2 - 2pz + p^2 \)[/tex]:
[tex]\[ z^2 - 10z + 25 = z^2 - 2 \cdot 5 \cdot z + 5^2 \][/tex]
So, you can see that [tex]\( p = 5 \)[/tex].
3. Write the factors: Since [tex]\( p = 5 \)[/tex], the polynomial can be factored as:
[tex]\[ (z - 5)^2 \][/tex]
So, the factored form of the polynomial [tex]\( z^2 - 10z + 25 \)[/tex] is:
[tex]\[ (z - 5)^2 \][/tex]
Therefore, the correct option from the given choices is:
[tex]\[ (z-5)(z-5) \][/tex]
1. Identify the polynomial structure: It is a quadratic polynomial of the form [tex]\( az^2 + bz + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 25 \)[/tex].
2. Check if it can be factored by recognizing perfect squares:
- A perfect square trinomial takes the form [tex]\( (z - p)^2 \)[/tex], which expands to [tex]\( z^2 - 2pz + p^2 \)[/tex].
- Here, compare the given polynomial [tex]\( z^2 - 10z + 25 \)[/tex] to [tex]\( z^2 - 2pz + p^2 \)[/tex]:
[tex]\[ z^2 - 10z + 25 = z^2 - 2 \cdot 5 \cdot z + 5^2 \][/tex]
So, you can see that [tex]\( p = 5 \)[/tex].
3. Write the factors: Since [tex]\( p = 5 \)[/tex], the polynomial can be factored as:
[tex]\[ (z - 5)^2 \][/tex]
So, the factored form of the polynomial [tex]\( z^2 - 10z + 25 \)[/tex] is:
[tex]\[ (z - 5)^2 \][/tex]
Therefore, the correct option from the given choices is:
[tex]\[ (z-5)(z-5) \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.