Question

(a+2b)(a^2+ab+b^2)(a-3b)(a^2+3ab+9b^2)​

Answer 1

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Answer 2

To simplify the expression $(a+2b)(a^2+ab+b^2)(a-3b)(a^2+3ab+9b^2)$, we can follow these steps:

Step 1: Recognize common terms and patterns.

The given expression consists of four separate factors. Let's label them as Factor 1, Factor 2, Factor 3, and Factor 4 for clarity:

Factor 1: $a + 2b$ Factor 2: $a^2 + ab + b^2$ Factor 3: $a - 3b$ Factor 4: $a^2 + 3ab + 9b^2$

Step 2: Group the factors with common terms.

Notice that Factor 2 is a sum of cubes, and Factor 4 is a sum of cubes as well. We can use the sum of cubes formula to simplify them:

Sum of cubes formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

So, Factor 2 can be rewritten as: $a^2 + ab + b^2 = (a + b)(a^2 - ab + b^2)$

And Factor 4 can be rewritten as: $a^2 + 3ab + 9b^2 = (a + 3b)(a^2 - 3ab + 9b^2)$

Step 3: Substitute the simplified factors back into the expression.

Now, we can rewrite the expression with the simplified factors:

$(a + 2b)(a + b)(a^2 - ab + b^2)(a - 3b)(a + 3b)(a^2 - 3ab + 9b^2)$

Step 4: Combine like terms.

Next, we can perform some simplifications by multiplying the terms with common factors:

$(a + 2b)(a + b)(a - 3b)(a + 3b)(a^2 - ab + b^2)(a^2 - 3ab + 9b^2)$

Step 5: Apply the difference of squares formula.

Notice that Factor 3 and Factor 4 are both difference of squares. The difference of squares formula is:

$a^2 - b^2 = (a + b)(a - b)$

We can use this formula to simplify the expression further:

$(a + 2b)(a + b)(a + 3b)(a - 3b)(a^2 - ab + b^2)(a^2 - 3ab + 9b^2)$

Step 6: Check for further simplification.

At this point, there are no more common terms that can be factored or simplified. So, the expression is now in its simplest form:

$(a + 2b)(a + b)(a + 3b)(a - 3b)(a^2 - ab + b^2)(a^2 - 3ab + 9b^2)$

This is the final simplified form of the given expression.