How to Factorize 278x3

Factorizing cubic expressions is a pivotal skill in algebra. One common method involves the sum of cubes formula, which can be applied in various scenarios. In this article, we will explore how to use the sum of cubes formula to factorize 278x3 by breaking down the process step by step.

Using the Sum of Cubes Formula

The sum of cubes formula states that:

a3 b3 (a b)(a2 - ab b2)

Given the expression 278x3, we can first rewrite it in a form that fits the sum of cubes formula. Let's separate the terms 27 and 8x3:

Step 1: Identifying Terms

27 8x3

We can identify a and b such that a3 27 and b3 8x3. From this, we find that:

a 3 b 2x

Now, we can apply the sum of cubes formula to this expression:

Step 2: Applying the Formula

27 8x3 33 (2x)3

Using the formula:

(3 2x)(32 - 3(2x) (2x)2)

Which simplifies to:

(3 2x)(9 - 6x 4x2)

Step 3: Final Factorization

Therefore, the factorized form of 278x3 is:

(3 2x)(9 - 6x 4x2)

Additional Factorization Examples

Example 1: x? - 8x3 27

Let's consider the expression x? - 8x3 27. This expression can be rewritten in a form that suggests the sum of cubes:

Step 1: Grouping Terms

x? - 8x3 27 x? - x327 27

Notice that -x327 27 resembles the sum of cubes formula. Let's set:

a2 x2 b2 3

Which gives:

a x b 3

Step 2: Applying the Formula

(x2)3 (-3)3

Using the sum of cubes formula:

(x2 3)((x2)2 - x2(-3) (-3)2)

Which simplifies to:

(x2 3)(x? 3x2 9)

Example 2: 8x3 - 27

Another example is the expression 8x3 - 27. We can directly apply the sum of cubes formula here:

Step 1: Identifying Terms

8x3 (2x)3 and 27 33

So, a 2x and b 3.

Step 2: Applying the Formula

(2x)3 - 33 (2x - 3)((2x)2 (2x)(3) 32)

Which simplifies to:

(2x - 3)(4x2 6x 9)

Conclusion

Understanding and applying the sum of cubes formula is crucial in simplifying and solving algebraic expressions. By breaking down the process step by step, we can easily factorize complex cubic expressions into simpler forms. Mastering this technique not only enhances problem-solving skills but also provides a deeper understanding of algebraic structures.

References

Algebraic Techniques for Factorization Sum of Cubes Formula in Algebra Examples of Factorizing Expressions