91Ó°ÊÓ

Expansion is an important concept in algebra. It involves rewriting an expression to show each term clearly. For example, the expansion of \( (x + 5)^3 + (x - 5)^3 \) means breaking down each term in the expression. To expand \( (x + 5)^3 \), we use the binomial theorem, and it becomes \( x^3 + 15x^2 + 75x + 125 \). Similarly, expanding \( (x - 5)^3 \), we get \( x^3 - 15x^2 + 75x - 125 \). These expanded terms can then be used to simplify the expression further. Remembering how to apply the binomial theorem is key in expanding expressions.

So, when we talk about expansion, we are essentially opening up the parentheses to see all the individual components that make up an expression.

Factoring is about breaking down complex expressions into simpler factors that, when multiplied together, give the original expression. For instance, in our example \( (x + 5)^3 + (x - 5)^3 \), instead of expanding directly, we use algebraic identities to factor it efficiently. One effective way to factor is recognizing common patterns, such as the sum of cubes. Knowing the sum of cubes formula is crucial here.

The sum of cubes formula is: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] This formula allows us to rewrite the expression \( (x+5)^3 + (x-5)^3 \) in a factored form without expanding.

Factoring helps simplify expressions and solve equations more easily. Learning different factoring techniques, like recognizing the difference of squares or the sum/product of roots, can make algebra much more manageable.

Algebraic identities are special equations that are true for all values of the variables involved. They are very useful in simplifying and solving algebraic expressions. In this exercise, we used one such identity: the sum of cubes. Using the sum of cubes formula, we directly factor \( (x + 5)^3 + (x - 5)^3 \) without expanding.

Let's rewrite the expression using the identity:
Let \( a = x + 5 \), and \( b = x - 5 \).
So, \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \).

Substituting back, we get:
\( a = x + 5 \) and \( b = x - 5 \) yields: \[ (x + 5 + x - 5)((x + 5)^2 - (x + 5)(x - 5) + (x - 5)^2) \] Simplifying further results in the expression: \[ 2x(2x^2 - 50) = 2x(2x^2 - 25) \] Understanding and using algebraic identities allows us to handle complex expressions efficiently. Practice makes perfect when it comes to recognizing and applying these identities in various problems.