3x^3-9ax^2-x+3a \\ 3x^3-9ax^2-
Answer (2)
Group the terms in the expression: ( 3 x 3 − 9 a x 2 ) + ( − x + 3 a ) .
Factor out common terms from each group: 3 x 2 ( x − 3 a ) − 1 ( x − 3 a ) .
Factor out the common binomial factor: ( 3 x 2 − 1 ) ( x − 3 a ) .
Further factor 3 x 2 − 1 as ( 3 x − 1 ) ( 3 x + 1 ) . The complete factorization is ( 3 x − 1 ) ( 3 x + 1 ) ( x − 3 a ) .
Explanation
Analyze the Expression We are given the expression 3 x 3 − 9 a x 2 − x + 3 a and asked to analyze it. The expression looks like it can be factored by grouping.
Group the Terms We group the terms as ( 3 x 3 − 9 a x 2 ) + ( − x + 3 a ) .
Factor out Common Terms We factor out common terms from each group: 3 x 2 ( x − 3 a ) − 1 ( x − 3 a ) .
Factor out the Common Binomial We factor out the common binomial factor ( x − 3 a ) to obtain ( 3 x 2 − 1 ) ( x − 3 a ) .
Further Analyze the Factor The expression is now factored as ( 3 x 2 − 1 ) ( x − 3 a ) . We can further analyze the factor 3 x 2 − 1 if needed, which can be written as ( 3 x − 1 ) ( 3 x + 1 ) .
Complete Factorization The complete factorization is ( 3 x − 1 ) ( 3 x + 1 ) ( x − 3 a ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used extensively in calculus and other advanced math courses. For example, factoring can help simplify complex expressions, solve equations, and analyze the behavior of functions. In real-world applications, factoring can be used in engineering to design structures, in physics to model motion, and in economics to analyze market trends. Understanding factoring helps in problem-solving across various fields.
The expression 3 x 3 − 9 a x 2 − x + 3 a can be factored by grouping and finding common factors, leading to the factorization ( 3 x − 1 ) ( 3 x + 1 ) ( x − 3 a ) . This technique is crucial in algebra and aids in simplifying complex expressions. Mastering factoring skills is beneficial for solving equations and further mathematical applications.
;
