$3 x^3-9 a x^2-x+3 a$
Answer (1)
Group the terms: ( 3 x 3 − 9 a x 2 ) + ( − x + 3 a ) .
Factor out common factors: 3 x 2 ( x − 3 a ) − 1 ( x − 3 a ) .
Factor out the common binomial: ( 3 x 2 − 1 ) ( x − 3 a ) .
Factor the difference of squares: ( 3 x − 1 ) ( 3 x + 1 ) ( x − 3 a ) .
Therefore, the factored expression is ( 3 x − 1 ) ( 3 x + 1 ) ( x − 3 a ) .
Explanation
Understanding the Problem We are given the expression 3 x 3 − 9 a x 2 − x + 3 a and we want to factor it.
Grouping Terms We can group the terms in pairs: ( 3 x 3 − 9 a x 2 ) + ( − x + 3 a ) .
Factoring Each Pair Now, we factor out the common factors from each pair: 3 x 2 ( x − 3 a ) − 1 ( x − 3 a ) .
Factoring out the Common Binomial We can see that ( x − 3 a ) is a common factor, so we factor it out: ( 3 x 2 − 1 ) ( x − 3 a ) .
Factoring as Difference of Squares We can further factor 3 x 2 − 1 as a difference of squares. Notice that 3 x 2 − 1 = ( s q r t 3 x ) 2 − 1 2 . Thus, we can factor it as ( s q r t 3 x − 1 ) ( s q r t 3 x + 1 ) .
Final Factored Expression Therefore, the fully factored expression is ( s q r t 3 x − 1 ) ( s q r t 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. Imagine you are designing a rectangular garden where the area is represented by the expression 3 x 3 − 9 a x 2 − x + 3 a . By factoring this expression, you can determine the possible dimensions (length and width) of the garden in terms of x and a . This allows you to plan the layout of the garden based on the available space and design constraints.
