What is $27 n^4+9 n^3$ in factored form?
Answer (2)
Find the greatest common factor (GCF) of the coefficients: GCF(27, 9) = 9.
Find the greatest common factor of the variable terms: GCF( n 4 , n 3 ) = n 3 .
Factor out the GCF 9 n 3 from the expression: 27 n 4 + 9 n 3 = 9 n 3 ( 3 n + 1 ) .
The factored form of the expression is 9 n 3 ( 3 n + 1 ) .
Explanation
Understanding the Problem We are asked to factor the expression 27 n 4 + 9 n 3 . This means we want to rewrite the expression as a product of simpler expressions.
Finding the GCF of the Coefficients First, we identify the greatest common factor (GCF) of the coefficients, 27 and 9. The GCF of 27 and 9 is 9, since 9 divides both 27 and 9.
Finding the GCF of the Variable Terms Next, we find the GCF of the variable terms, n 4 and n 3 . The GCF of n 4 and n 3 is n 3 , since n 3 is the highest power of n that divides both n 4 and n 3 .
Factoring out the GCF Now, we factor out the GCF, which is 9 n 3 , from the expression 27 n 4 + 9 n 3 . We have:
27 n 4 + 9 n 3 = 9 n 3 ( 3 n ) + 9 n 3 ( 1 )
Writing the Factored Form Finally, we write the expression in factored form by factoring out 9 n 3 :
27 n 4 + 9 n 3 = 9 n 3 ( 3 n + 1 )
Final Answer Therefore, the factored form of 27 n 4 + 9 n 3 is 9 n 3 ( 3 n + 1 ) .
Examples
Factoring is a fundamental skill in algebra that helps simplify complex expressions and solve equations. For instance, if you're designing a rectangular garden where the area is represented by the expression 27 n 4 + 9 n 3 , factoring it into 9 n 3 ( 3 n + 1 ) allows you to easily determine possible dimensions for the garden. If n represents a basic unit of length, you can quickly find suitable values for n that make practical sense for the garden's layout, ensuring efficient use of space and resources.
The expression 27 n 4 + 9 n 3 can be factored as 9 n 3 ( 3 n + 1 ) by identifying the greatest common factor of the coefficients and variable parts. The GCF for the coefficients is 9, and for the variable parts, it is n 3 . Thus, the final factored form is obtained by rewriting the expression with the GCF taken out.
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