Problem 54 - Algebra A Combined Function

Question

Subtract \(\left(4 x^{2}-2 x+2\right)\) from the sum of \(\left(x^{2}+7 x+1\right)\) and \((7 x+5)\)

Step-by-Step Solution

Answer

The result is \(-3x^2 + 16x + 4\).

Step 1: Find the Sum

First, we need to find the sum of the expressions \((x^2 + 7x + 1)\) and \((7x + 5)\). Add the like terms together: \(x^2 + 7x + 7x + 1 + 5\). This simplifies to \(x^2 + 14x + 6\).

Step 2: Set Up the Subtraction

We need to subtract \((4x^2 - 2x + 2)\) from the result of the previous step, \(x^2 + 14x + 6\). This is written as \((x^2 + 14x + 6) - (4x^2 - 2x + 2)\).

Step 3: Distribute the Negative

Distribute the negative sign through the second polynomial: \(x^2 + 14x + 6 - 4x^2 + 2x - 2\). This changes the sign of each term in \(4x^2 - 2x + 2\).

Step 4: Combine Like Terms

Combine the like terms from the expression \(x^2 + 14x + 6 - 4x^2 + 2x - 2\):- Combine the \(x^2\) terms: \(x^2 - 4x^2 = -3x^2\).- Combine the \(x\) terms: \(14x + 2x = 16x\).- Combine the constant terms: \(6 - 2 = 4\).The result is \(-3x^2 + 16x + 4\).

Key Concepts

Polynomial Addition

When we talk about polynomial addition, we mean putting together terms that have the same variables and exponents. It’s like adding apples with apples and oranges with oranges.
For example, if we have two polynomials, \(x^2 + 7x + 1\) and \(7x + 5\), we add them by finding terms with the same variables: \(x^2\), \(x\), and constant numbers.
This means our first step is to combine the respective terms: \(x^2 + 7x + 1 + 7x + 5\). It simplifies by adding the coefficients (numbers in front) of the like terms: \(x^2\), \(7x + 7x = 14x\), and \(1 + 5 = 6\).

The result is \(x^2 + 14x + 6\). It’s important because putting things together correctly sets the stage for accurate subtraction.

Combining Like Terms

Combining like terms in polynomial arithmetic involves simplifying expressions by merging terms with the same variable raised to the same power. Let’s think about it as gathering all "x" values and similar kinds of numbers together.
In the example with \(x^2 + 14x + 6 - 4x^2 + 2x - 2\), we need to identify the groups of like terms. That includes matching up x-squared terms, x terms, and constant terms.
Here’s how it breaks down:

• The x-squared terms: \(x^2 - 4x^2\) result in \(-3x^2\).
• The x-terms: \(14x + 2x\) result in \(16x\).
• The constants: \(6 - 2\) result in \(4\).

By combining like terms, we simplify the polynomials making them easier to work with, and ultimately, resulting in \(-3x^2 + 16x + 4\). This keeps our work organized and neat.

Distributive Property

The distributive property is a crucial concept in algebra used to simplify expressions, especially in operations such as subtraction. When you subtract one polynomial from another, you often distribute a negative sign across all terms of the second polynomial.
This step is essential. It changes the sign of the terms in the polynomial being subtracted. For instance, in \(x^2 + 14x + 6 - (4x^2 - 2x + 2)\), distributing the negative sign results in changing signs to \(-4x^2 + 2x - 2\).

Without this step, you might forget to flip the signs, leading to incorrect calculations. So remember, when you see \( - (a + b + c)\), think of it as \(-a - b - c\).

• This property applies not only in subtraction but also in expanding expressions like \(a(b + c) = ab + ac\).

Understanding and using the distributive property ensures accuracy in solving polynomial expressions.