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Answer :
To simplify the expression [tex]\(3n^4 - 147\)[/tex], we'll look for any possible simplifications or factorizations.
1. Identify Common Factors: Check if there are any common factors between the terms in the expression. In this case, [tex]\(3n^4\)[/tex] and 147 both have a common factor of 3.
2. Factor Out the Common Factor: When you factor out the common factor of 3, you'll get:
[tex]\[
3(n^4 - 49)
\][/tex]
3. Simplify Further Using Difference of Squares:
- Notice that [tex]\(n^4 - 49\)[/tex] is a difference of squares. Recall that a difference of squares is expressed as [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
- Here, [tex]\(n^4\)[/tex] is [tex]\((n^2)^2\)[/tex] and 49 is [tex]\(7^2\)[/tex].
- Therefore, you can factor [tex]\(n^4 - 49\)[/tex] as:
[tex]\[
(n^2 - 7)(n^2 + 7)
\][/tex]
4. Write the Fully Factored Expression:
- Substitute back the factored form into the expression with the common factor:
[tex]\[
3(n^2 - 7)(n^2 + 7)
\][/tex]
This is the simplified, factored form of the expression [tex]\(3n^4 - 147\)[/tex].
1. Identify Common Factors: Check if there are any common factors between the terms in the expression. In this case, [tex]\(3n^4\)[/tex] and 147 both have a common factor of 3.
2. Factor Out the Common Factor: When you factor out the common factor of 3, you'll get:
[tex]\[
3(n^4 - 49)
\][/tex]
3. Simplify Further Using Difference of Squares:
- Notice that [tex]\(n^4 - 49\)[/tex] is a difference of squares. Recall that a difference of squares is expressed as [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
- Here, [tex]\(n^4\)[/tex] is [tex]\((n^2)^2\)[/tex] and 49 is [tex]\(7^2\)[/tex].
- Therefore, you can factor [tex]\(n^4 - 49\)[/tex] as:
[tex]\[
(n^2 - 7)(n^2 + 7)
\][/tex]
4. Write the Fully Factored Expression:
- Substitute back the factored form into the expression with the common factor:
[tex]\[
3(n^2 - 7)(n^2 + 7)
\][/tex]
This is the simplified, factored form of the expression [tex]\(3n^4 - 147\)[/tex].
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