Problem 57 - Elementary and Intermediate Algebra

Question

Use vertical form to subtract the polynomials. $$ \text { Subtract } 2 x^{2}-2 x+3 \text { from } 3 x^{2}+4 x+5 $$

Step-by-Step Solution

Answer

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

Step 1: Align the Polynomials

We start by writing the two polynomials in a vertical format. Align similar terms one on top of the other: \[ \begin{array}{r} 3x^2 + 4x + 5 \-\ (2x^2 - 2x + 3) \end{array} \].

Step 2: Distribute the Negative Sign

Subtracting a polynomial means we add the opposite. Distribute the negative sign across the second polynomial:\[ -(2x^2 - 2x + 3) = -2x^2 + 2x - 3 \].

Step 3: Combine Like Terms

We now perform the operation on each set of like terms:- For \(x^2\) terms: \(3x^2 - 2x^2 = 1x^2\).- For \(x\) terms: \(4x + 2x = 6x\).- For constants: \(5 - 3 = 2\).Put them all together: \[ 1x^2 + 6x + 2 \].

Step 4: Simplify if Necessary

Check to see if any further simplification is possible. The polynomial \(1x^2 + 6x + 2\) is fully simplified.

Key Concepts

Vertical Form

When subtracting polynomials, placing them in a vertical form is a highly effective strategy. This visual alignment helps in systematically arranging the polynomials. By placing one directly above the other, you align similar terms—making it easy to see which elements to subtract or add. This setup is similar to addition and subtraction done in arithmetic, where you stack numbers vertically.

• Write each polynomial on a separate line.
• Ensure terms of the same degree (like the terms with the same variables and exponents) are aligned.
• Align constant terms as well.

By following this setup, the process becomes much more organized. For our example, we line up the polynomials as follows:\[\begin{array}{r}3x^2 + 4x + 5 \\-(2x^2 - 2x + 3)\end{array}\]After this, you can easily work with each column to perform the subtraction.

Like Terms

"Like terms" are a fundamental concept when working with polynomials. These are terms within an expression that have the same variable and the same exponent. When performing operations like addition or subtraction of polynomials, you can only combine like terms. This step simplifies the expression and makes the polynomial easier to manage.

• For instance, in the polynomial \(3x^2 + 4x + 5\), \(3x^2\) is a like term with \(2x^2\) because they both involve \(x^2\).
• Similarly, the terms \(4x\) and \(-2x\) are like terms because they both involve \(x\).
• Constant terms like \(5\) and \(3\) are grouped because they do not involve any variables.

In our example, after distributing the negative sign (which we discuss next), we combine like terms as follows: \(3x^2 - 2x^2 = 1x^2\), \(4x + 2x = 6x\), and \(5 - 3 = 2\). This reduces the polynomial to a simpler form.

Distributing Negative Sign

A crucial step in polynomial subtraction is distributing the negative sign across the terms of the polynomial being subtracted. This involves flipping the sign of each term in the second polynomial. Think of it as multiplying the entire polynomial by negative one.

• Apply the negative sign to each term individually.
• For example, for the polynomial \(2x^2 - 2x + 3\), applying the negative sign gives \(-2x^2 + 2x - 3\).
• This transforms the subtraction into an addition of the opposite polynomial.

Once this is done, you are ready to methodically combine like terms, making sure to carefully switch each sign before doing any calculations. In our example, the transformed polynomial is then easily combined with the terms of the first polynomial, leading to a simplified result of \(1x^2 + 6x + 2\). Taking this step prevents mistakes and ensures the accuracy of your solution during subtraction.