Problem 25 - Algebra for College Students

Question

Subtract the polynomials using the horizontal format. \(3 x^{2}-x+2\) from \(7 x^{2}+9 x+8\)

Step-by-Step Solution

Answer

The result is \(4x^2 + 10x + 6\).

Step 1: Write Down the Polynomials

We need to subtract the polynomial \(3x^2 - x + 2\) from \(7x^2 + 9x + 8\). Write them both down in the horizontal format: \((7x^2 + 9x + 8) - (3x^2 - x + 2)\).

Step 2: Distribute the Negative Sign

Apply the negative sign to each term in the polynomial being subtracted. This changes each term to its additive inverse: \((7x^2 + 9x + 8) - 3x^2 + x - 2\).

Step 3: Combine Like Terms

Combine the like terms from both polynomials: \[7x^2 - 3x^2 = 4x^2\]\[9x + x = 10x\]\[8 - 2 = 6\]The result is \(4x^2 + 10x + 6\).

Step 4: Final Expression

The result of subtracting the polynomial \(3x^2 - x + 2\) from \(7x^2 + 9x + 8\) is \(4x^2 + 10x + 6\).

Key Concepts

Combining Like Terms

Combining like terms is a crucial step when working with polynomials. When you perform operations like addition or subtraction of polynomials, you can only combine the coefficients of terms with the same variable raised to the same power. This means that terms like \(3x^2\) and \(7x^2\) can be combined because they both have the variable \(x\) raised to the power of 2. However, terms like \(3x^2\) and \(9x\) cannot be directly combined because they do not share the same power of the variable.

Here’s how it works in practice:

• Identify terms with the same variables and exponents.
• Add or subtract their coefficients.

For example, in our exercise, we combine like terms to simplify the expression:

• \(7x^2 - 3x^2\) results in \(4x^2\)
• \(9x + x\) results in \(10x\)
• \(8 - 2\) results in \(6\).

This process of combining like terms simplifies the polynomial to make it easier to understand and work with.

Additive Inverse

Before subtracting polynomials, you must understand the concept of the additive inverse. The additive inverse of a number is simply the number that, when added to the original number, equals zero. When dealing with polynomials, each term within the polynomial has an additive inverse.

For example, consider the polynomial we are subtracting: \(3x^2 - x + 2\). The additive inverse of each term is:

• The inverse of \(3x^2\) is \(-3x^2\).
• The inverse of \(-x\) is \(x\).
• The inverse of \(2\) is \(-2\).

Applying these inverses is like distributing a negative sign across the polynomial you are subtracting, which transforms it into addition. This step is essential because it sets the stage for successfully combining like terms in the next step. Without creating the additive inverse, we could not correctly perform the subtraction of the polynomials.

Polynomial Operations

Polynomial operations like addition and subtraction are fundamental algebraic processes that follow specific rules similar to arithmetic. When performing polynomial operations, keeping the terms organized and following steps carefully will lead to successful results.

Here is the process we went through in solving the exercise:

• First, express the subtraction in a horizontal format.
• Second, apply the additive inverse to transform subtraction into addition.
• Third, combine like terms to simplify the polynomial.

Polynomial subtraction, much like addition, involves manipulating terms with respect to their variables and coefficients. Ensuring that like terms are properly dealt with and the additive inverses are correctly applied are key to simplifying the result. Understanding these basic polynomial operations is not only helpful for this exercise but lays the groundwork for more complicated algebraic manipulations you will encounter in advanced mathematics.