Simplifying Polynomial Expressions: A^2 - (B+C)

When diving into the world of algebra, you'll often encounter situations where you need to simplify complex expressions involving polynomials. One such challenge is to determine $A^2 - (B+C)$ in its simplest form, given specific polynomial definitions for $A$, $B$, and $C$. This process requires a systematic approach to ensure accuracy. We'll break down each step, from squaring polynomial $A$ to subtracting the sum of polynomials $B$ and $C$. Understanding how to manipulate these algebraic expressions is a foundational skill that opens doors to solving more intricate mathematical problems. Let's get started on unraveling this polynomial puzzle!

Understanding the Polynomials

Before we begin the simplification, let's clearly define the polynomials we're working with. We have:

• A = 3x - 4
• B = x + 7
• C = x² + 2

Our goal is to calculate $A^2 - (B+C)$ and express the result in its simplest form. This means combining like terms and presenting the polynomial in standard form, typically with the highest power of $x$ first.

Step 1: Calculating $A^2$

The first part of our expression is $A^2$. This means we need to square the polynomial $A$. Since $A = 3x - 4$, we will calculate $(3x - 4)^2$. To do this, we can use the formula for squaring a binomial, which is $(a-b)^2 = a^2 - 2ab + b^2$. In our case, $a = 3x$ and $b = 4$.

Applying the formula:

$A^2 = (3x - 4)^2$

$A^2 = (3x)^2 - 2(3x)(4) + (4)^2$

$A^2 = 9x^2 - 24x + 16$

So, the square of polynomial $A$ is $9x^2 - 24x + 16$. It's crucial to perform this step accurately, as any errors here will propagate through the rest of the calculation. Remember to square both the coefficient and the variable when squaring $3x$, and don't forget the middle term, which comes from multiplying $2$, $3x$, and $4$. The final term is simply the square of the constant, $4$. This expansion gives us a quadratic polynomial.

Step 2: Calculating $B+C$

Next, we need to find the sum of polynomials $B$ and $C$. We have $B = x + 7$ and $C = x^2 + 2$. To add them, we simply combine like terms:

$B+C = (x + 7) + (x^2 + 2)$

$B+C = x^2 + x + (7 + 2)$

$B+C = x^2 + x + 9$

Here, we identified that the only $x^2$ term is from $C$, the only $x$ term is from $B$, and the constants $7$ and $2$ from $B$ and $C$ respectively are combined to form $9$. This addition results in another quadratic polynomial, $x^2 + x + 9$. This step is generally more straightforward than squaring a binomial, but it still requires careful attention to identify and combine terms with the same power of $x$.

Step 3: Calculating $A^2 - (B+C)$

Now we have all the pieces to calculate the final expression: $A^2 - (B+C)$. We substitute the results from Step 1 and Step 2:

$A^2 - (B+C) = (9x^2 - 24x + 16) - (x^2 + x + 9)$

When subtracting polynomials, it's essential to distribute the negative sign to every term in the polynomial being subtracted. This is a common point where errors can occur.

$A^2 - (B+C) = 9x^2 - 24x + 16 - x^2 - x - 9$

Now, we group and combine like terms:

• $x^2$ terms: $9x^2 - x^2 = 8x^2$
• $x$ terms: $-24x - x = -25x$
• Constant terms: $16 - 9 = 7$

Putting it all together, we get:

$A^2 - (B+C) = 8x^2 - 25x + 7$

This is the simplest form of the expression. We have combined all like terms, resulting in a single quadratic polynomial. The standard form arranges the terms from highest to lowest power of $x$. In this case, the order is already correct: $x^2$, then $x$, then the constant term. This final expression, $8x^2 - 25x + 7$, represents the solution in its most concise and simplified form.

Conclusion

By following these methodical steps, we've successfully simplified the expression $A^2 - (B+C)$ to $8x^2 - 25x + 7$. The process involved squaring a binomial and subtracting polynomials, both fundamental operations in algebra. Mastering these techniques is key to tackling more complex algebraic challenges.

For those interested in further exploring the properties and operations of polynomials, a great resource is the Khan Academy website. They offer comprehensive lessons and practice exercises on various mathematics topics, including advanced algebra. You can find detailed explanations on polynomial manipulation at Khan Academy Mathematics.