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 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. To do this, we will identify the greatest common factor (GCF) of the terms and factor it out.
Finding the GCF of the Coefficients First, let's find the GCF of the coefficients, 27 and 9. The greatest common factor of 27 and 9 is 9, since 9 divides both 27 and 9, and no larger number does. We can write 27 = 9 × 3 and 9 = 9 × 1 .
Finding the GCF of the Variable Terms Next, let's find the GCF of the variable terms, n 4 and n 3 . The greatest common factor of n 4 and n 3 is n 3 , since n 3 divides both n 4 and n 3 , and no higher power of n does. We can write n 4 = n 3 × n and n 3 = n 3 × 1 .
Factoring out the GCF Now, we can 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 ) = 9 n 3 ( 3 n + 1 ) So, the factored form of the expression is 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 and is used extensively in solving equations and simplifying expressions. For example, if you want to find the roots of the polynomial 27 n 4 + 9 n 3 = 0 , factoring it as 9 n 3 ( 3 n + 1 ) = 0 makes it easy to see that the roots are n = 0 (with multiplicity 3) and n = − 3 1 . Factoring also helps in simplifying complex rational expressions, making them easier to work with in calculus and other advanced math courses.
The expression 27 n 4 + 9 n 3 can be factored as 9 n 3 ( 3 n + 1 ) by first finding the greatest common factor of the coefficients and the variable parts. This GCF is 9 n 3 , which we can factor out. The final result is 9 n 3 ( 3 n + 1 ) .
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