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Answer :
Sure! Let's factor the polynomial [tex]\(80z^4 - 3125\)[/tex] step by step:
1. Recognize the Difference of Powers:
The expression [tex]\(80z^4 - 3125\)[/tex] can be viewed as a difference of terms involving powers of [tex]\(z\)[/tex] and constants:
[tex]\[
80z^4 - 3125 = 16z^4 - 5^5
\][/tex]
This allows you to identify any common factors and establish a difference of squares or related forms.
2. Factor Out the Greatest Common Factor (GCF):
Notice that both terms don't have any “simple” common factors (like 1, 2, etc.). However, if there are none, we can proceed directly to see if we can apply principles like a difference of squares or sum/difference of cubes.
3. Factor Using Difference of Squares:
Recognizing that we have a difference here, we express [tex]\(80z^4\)[/tex] as [tex]\((4z^2)^2\)[/tex] and [tex]\(3125\)[/tex] as [tex]\(5^5\)[/tex]. The expression becomes:
[tex]\[
(4z^2)^2 - 5^2 \times (5)^2
\][/tex]
This allows us to factor by recognizing patterns:
- [tex]\( (a^2 - b^2) = (a-b)(a+b) \)[/tex]
- Here, treat [tex]\(a = 4z^2\)[/tex] and [tex]\(b^2 = 5^4\)[/tex]. Simplifying the powers in terms of factors, we now create expressions in expanded polynomial terms:
4. Implement Binomial Expansions as Needed:
Using the apparent perfect squares or simplest means to factor based on previous calculations:
[tex]\[
80z^4 - 3125 = 5 \times (16z^4 - 625)
\][/tex]
- Here, continue factoring:
- [tex]\(16z^4 - 625 = (4z^2 - 25)(4z^2 + 25)\)[/tex]
- Notice [tex]\(4z^2 - 25 = (2z - 5)(2z + 5)\)[/tex]
5. Combine the Results:
Therefore, by using the formula and reasoning, you get:
[tex]\[
80z^4 - 3125 = 5 \times (2z - 5) \times (2z + 5) \times (4z^2 + 25)
\][/tex]
So, the fully factored form of the polynomial [tex]\(80z^4 - 3125\)[/tex] is:
[tex]\[
5 \times (2z - 5) \times (2z + 5) \times (4z^2 + 25)
\][/tex]
These steps illustrate the structured approach to going from an initial polynomial expression to its factored form.
1. Recognize the Difference of Powers:
The expression [tex]\(80z^4 - 3125\)[/tex] can be viewed as a difference of terms involving powers of [tex]\(z\)[/tex] and constants:
[tex]\[
80z^4 - 3125 = 16z^4 - 5^5
\][/tex]
This allows you to identify any common factors and establish a difference of squares or related forms.
2. Factor Out the Greatest Common Factor (GCF):
Notice that both terms don't have any “simple” common factors (like 1, 2, etc.). However, if there are none, we can proceed directly to see if we can apply principles like a difference of squares or sum/difference of cubes.
3. Factor Using Difference of Squares:
Recognizing that we have a difference here, we express [tex]\(80z^4\)[/tex] as [tex]\((4z^2)^2\)[/tex] and [tex]\(3125\)[/tex] as [tex]\(5^5\)[/tex]. The expression becomes:
[tex]\[
(4z^2)^2 - 5^2 \times (5)^2
\][/tex]
This allows us to factor by recognizing patterns:
- [tex]\( (a^2 - b^2) = (a-b)(a+b) \)[/tex]
- Here, treat [tex]\(a = 4z^2\)[/tex] and [tex]\(b^2 = 5^4\)[/tex]. Simplifying the powers in terms of factors, we now create expressions in expanded polynomial terms:
4. Implement Binomial Expansions as Needed:
Using the apparent perfect squares or simplest means to factor based on previous calculations:
[tex]\[
80z^4 - 3125 = 5 \times (16z^4 - 625)
\][/tex]
- Here, continue factoring:
- [tex]\(16z^4 - 625 = (4z^2 - 25)(4z^2 + 25)\)[/tex]
- Notice [tex]\(4z^2 - 25 = (2z - 5)(2z + 5)\)[/tex]
5. Combine the Results:
Therefore, by using the formula and reasoning, you get:
[tex]\[
80z^4 - 3125 = 5 \times (2z - 5) \times (2z + 5) \times (4z^2 + 25)
\][/tex]
So, the fully factored form of the polynomial [tex]\(80z^4 - 3125\)[/tex] is:
[tex]\[
5 \times (2z - 5) \times (2z + 5) \times (4z^2 + 25)
\][/tex]
These steps illustrate the structured approach to going from an initial polynomial expression to its factored form.
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