Answer :
To factor the trinomial [tex]\(3n^2 - 42n + 147\)[/tex] completely, follow these steps:
1. Identify the Common Factor:
First, notice that all the terms in the trinomial [tex]\(3n^2 - 42n + 147\)[/tex] have a common factor. We can factor out a 3 from each term:
[tex]\[
3(n^2 - 14n + 49)
\][/tex]
2. Factor the Quadratic:
Next, let's focus on the quadratic inside the parentheses, [tex]\(n^2 - 14n + 49\)[/tex]. We need to factor this expression.
3. Recognize a Perfect Square:
Notice that [tex]\(n^2 - 14n + 49\)[/tex] is a perfect square trinomial. It can be expressed as [tex]\((n - 7)^2\)[/tex] because:
- The square root of the first term, [tex]\(n^2\)[/tex], is [tex]\(n\)[/tex].
- The square root of the last term, [tex]\(49\)[/tex], is 7.
- When doubled, the product of these terms gives the middle term: [tex]\(2 \times n \times 7 = 14n\)[/tex].
4. Write the Factorized Form:
Therefore, the expression [tex]\(n^2 - 14n + 49\)[/tex] factors to [tex]\((n - 7)^2\)[/tex]. Putting it all together, we have:
[tex]\[
3(n - 7)^2
\][/tex]
So the complete factorization of the trinomial is [tex]\(\boxed{3(n - 7)^2}\)[/tex]. Thus, option A is the correct choice.
1. Identify the Common Factor:
First, notice that all the terms in the trinomial [tex]\(3n^2 - 42n + 147\)[/tex] have a common factor. We can factor out a 3 from each term:
[tex]\[
3(n^2 - 14n + 49)
\][/tex]
2. Factor the Quadratic:
Next, let's focus on the quadratic inside the parentheses, [tex]\(n^2 - 14n + 49\)[/tex]. We need to factor this expression.
3. Recognize a Perfect Square:
Notice that [tex]\(n^2 - 14n + 49\)[/tex] is a perfect square trinomial. It can be expressed as [tex]\((n - 7)^2\)[/tex] because:
- The square root of the first term, [tex]\(n^2\)[/tex], is [tex]\(n\)[/tex].
- The square root of the last term, [tex]\(49\)[/tex], is 7.
- When doubled, the product of these terms gives the middle term: [tex]\(2 \times n \times 7 = 14n\)[/tex].
4. Write the Factorized Form:
Therefore, the expression [tex]\(n^2 - 14n + 49\)[/tex] factors to [tex]\((n - 7)^2\)[/tex]. Putting it all together, we have:
[tex]\[
3(n - 7)^2
\][/tex]
So the complete factorization of the trinomial is [tex]\(\boxed{3(n - 7)^2}\)[/tex]. Thus, option A is the correct choice.
