Understanding the Expression to Factorize

We are asked to factorize the expression: \(x^2 - y^2 - 9z^2 + 6yz\).

Factorization involves rewriting an expression as a product of its factors. To factorize this specific expression, we need to look for patterns or identities that we can use. We have four terms, and they involve squares and products of variables \(x\), \(y\), and \(z\). This suggests we might be able to use algebraic identities like the difference of squares (\(a^2 - b^2\)) or the square of a binomial (\((a \pm b)^2\)).

Rearranging Terms for Factorization

Let's look at the terms involving \(y\) and \(z\): \(-y^2 - 9z^2 + 6yz\). It's often helpful to group terms that look like they might form a perfect square. The terms \(y^2\), \(9z^2\), and \(6yz\) seem related to the expansion of a binomial squared. Notice that \(9z^2 = (3z)^2\).

Let's rearrange the expression to group the terms involving \(y\) and \(z\):

\(x^2 - (y^2 + 9z^2 - 6yz)\)

We factored out a negative sign from the last three terms. Now, let's examine the expression inside the parenthesis: \(y^2 + 9z^2 - 6yz\). This looks very similar to the expansion of \((a-b)^2 = a^2 - 2ab + b^2\).

Recognizing the Perfect Square Trinomial

Let's compare \(y^2 + 9z^2 - 6yz\) with \(a^2 - 2ab + b^2\):

• \(a^2 = y^2 \implies a = y\)
• \(b^2 = 9z^2 \implies b = 3z\)
• Check the middle term: \(2ab = 2(y)(3z) = 6yz\).

Since the middle term in our expression is \(-6yz\), the terms inside the parenthesis must be the expansion of \((y - 3z)^2\) or \((3z - y)^2\). Let's verify:

• \((y - 3z)^2 = y^2 - 2(y)(3z) + (3z)^2 = y^2 - 6yz + 9z^2\). This matches the expression inside the parenthesis.

So, the original expression can be rewritten as:

\(x^2 - (y - 3z)^2\)

Applying the Difference of Squares Formula

The expression is now in the form of \(a^2 - b^2\), which is the difference of squares. The formula for the difference of squares is \(a^2 - b^2 = (a - b)(a + b)\).

In our expression \(x^2 - (y - 3z)^2\):

• \(a = x\)
• \(b = (y - 3z)\)

Now, we apply the formula:

• \(a - b = x - (y - 3z) = x - y + 3z\)
• \(a + b = x + (y - 3z) = x + y - 3z\)

Therefore, the factorization of the expression is the product of \((a-b)\) and \((a+b)\).

\(x^2 - (y - 3z)^2 = (x - (y - 3z))(x + (y - 3z))\)

\( = (x - y + 3z)(x + y - 3z)\)

Final Factored Form

The factored form of the expression \(x^2 - y^2 - 9z^2 + 6yz\) is \((x - y + 3z)(x + y - 3z)\). This shows how to successfully factorize a complex expression by recognizing common algebraic patterns. To fully factorize means breaking it down into simpler expressions that multiply together. We used the perfect square identity first, and then the difference of squares identity to factorize completely.