Problem 44 - Algebra for College Students

Question

Perform the operations as described. Subtract \(-4 x^{2}+6 x-3\) from the sum of \(-3 x+4\) and \(9 x^{2}-6\)

Step-by-Step Solution

Answer

13x^2 - 9x + 1

Step 1: Finding the Sum

First, find the sum of the expressions -3x + 4 and 9x^2 - 6. Combine them to get: (9x^2 - 6) + (-3x + 4). Rewriting gives: 9x^2 - 3x + 4 - 6. Now, simplify by combining like terms. Two constants, 4 and -6, combine to give -2. Thus, the sum is 9x^2 - 3x - 2.

Step 2: Subtracting the Polynomial

We need to subtract the polynomial (-4x^2 + 6x - 3) from the sum obtained in Step 1. This means we have: (9x^2 - 3x - 2) - (-4x^2 + 6x - 3). Distribute the negative across the terms in the second polynomial: 9x^2 - 3x - 2 + 4x^2 - 6x + 3.

Step 3: Combining Like Terms

Now, combine the like terms from the expression obtained: (9x^2 + 4x^2) yields 13x^2, (-3x - 6x) yields -9x, and (-2 + 3) yields 1. Therefore, the final result after subtracting is: 13x^2 - 9x + 1.

Key Concepts

Combining Like Terms

Combining like terms is a key step when working with polynomials. A polynomial consists of terms, and each term is usually a product of a number and a variable raised to a power.
Like terms are terms that have the same variable parts, meaning the same variables raised to the same powers. For example:

• In the expression \(9x^2 - 3x + 4x^2 + 4\), \(9x^2\) and \(4x^2\) are like terms because they both contain \(x^2\).
• \(-3x\) is not a like term with the \(x^2\) terms since it contains \(x\), not \(x^2\).
• The constants \(+4\) and \(-6\) are like terms because they are numbers without variables.

Once identified, simply add or subtract the coefficients of these terms. In our example, combining \(9x^2\) and \(4x^2\) yields \(13x^2\), while combining the constants \(+4\) and \(-6\) yields \(-2\).
Combining like terms simplifies expressions and is essential for further operations such as subtraction or graphical representation of the polynomial.

Subtracting Polynomials

Subtracting polynomials requires careful attention to signs. When we subtract polynomials, it's like distributing a "negative one" across the entire polynomial we are subtracting.
For example, if we have two polynomials \(A - B\), and \(A = 9x^2 - 3x - 2\) and \(B = -4x^2 + 6x - 3\), then subtracting \(B\) from \(A\) requires us to subtract each term of \(B\):

• Change each positive term in \(B\) to negative.
• Change each negative term in \(B\) to positive.

So in this case:

• \(-(-4x^2)\) becomes \(+4x^2\)
• \(-(+6x)\) becomes \(-6x\)
• \(-(-3)\) becomes \(+3\)

This changes the expression from \(9x^2 - 3x - 2 - (-4x^2 + 6x - 3)\) to \(9x^2 - 3x - 2 + 4x^2 - 6x + 3\).
After changing the signs of \(B\), you can then combine like terms. This ensures the subtraction is performed correctly and we achieve the correct result.

Simplifying Expressions

Simplifying expressions is all about making an expression as concise as possible. This involves combining like terms and performing arithmetic on numbers.
After subtracting polynomials, the expression \((9x^2 + 4x^2) - (3x + 6x) - 2 + 3\) is obtained.
Our goal is to present this expression in its simplest form possible. Here's how to do it:

• Begin by adding and subtracting the coefficients of like terms: \(9x^2 + 4x^2\) gives \(13x^2\), and \(-3x - 6x\) adds up to \(-9x\).
• The numerical constants \(-2\) and \(+3\) simplify to \(+1\).

At this point, your simplified polynomial is \(13x^2 - 9x + 1\).
A simplified expression is easier to understand, analyze, and work with. It forms the basis for various mathematical operations such as finding roots or graphing. Always remember: simplifying is not just an end-stage task but a crucial part in every arithmetic or algebraic process.