Factor (3x+y)^2-9
The question provided is asking for the factorization of a given algebraic expression. Specifically, it's requesting to express the difference of squares, (3x + y)^2 - 9, as a product of two binomial expressions. In other words, you should apply the difference of squares formula, which states that a^2 - b^2 can be factored into (a + b)(a - b), to rewrite the given expression in a factored form that is easier to work with or understand.
$\left(\left(\right. 3 x + y \left.\right)\right)^{2} - 9$
Answer
Solution:
Step 1:
Express the number $9$ as the square of $3$, which is $3^2$. Thus, the expression becomes $(3x+y)^2 - 3^2$.
Step 2:
Recognize that we have a difference of squares, which can be factored according to the identity $a^2 - b^2 = (a + b)(a - b)$. Here, $a = 3x + y$ and $b = 3$. Applying this identity, we get $(3x+y+3)(3x+y-3)$.
Knowledge Notes:
The problem involves factoring a quadratic expression that is in the form of a difference of squares. The difference of squares is a mathematical pattern where the difference between two perfect square terms can be factored into the product of the sum and difference of the square roots of those terms. The general formula for the difference of squares is:
$$ a^2 - b^2 = (a + b)(a - b) $$
In this formula, $a$ and $b$ are any expressions where $a^2$ and $b^2$ are perfect squares. When factoring an expression using the difference of squares formula, it is important to identify the correct $a$ and $b$ terms. In the given problem, $a$ is identified as $(3x + y)$ and $b$ is identified as $3$ since $3^2 = 9$.
This factoring technique is commonly used in algebra and is an essential tool for simplifying expressions and solving equations. It is particularly useful when dealing with quadratic expressions and polynomials. Understanding the difference of squares and other factoring techniques is crucial for solving a wide range of mathematical problems.
