Problem 60 - Elementary and Intermediate Algebra

Question

Use vertical form to subtract the polynomials. $$ \text { Subtract }\left(4 p^{2}-4 p-40\right) \text { from }\left(10 p^{2}-30\right) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the polynomials

The polynomials involved are \(10p^2 - 30\) (the minuend) and \(4p^2 - 4p - 40\) (the subtrahend). We need to subtract the second polynomial from the first.

Step 2: Align the polynomials vertically

Write the polynomials in vertical form, aligning like terms vertically. Ensure you account for all types of terms (i.e., \(p^2\), \(p\), and constants). If a term is missing, remember to place a zero as a placeholder: \[ \begin{array}{r} 10p^2 +0p - 30 \ -(4p^2 - 4p - 40) \ \hline \end{array} \]

Step 3: Subtract the polynomials term by term

Subtract the coefficients of like terms starting from the highest degree:1. Subtract the \(p^2\) terms: \(10p^2 - 4p^2 = 6p^2\).2. Subtract the \(p\) terms: \(0p - (-4p) = 4p\) (subtracting a negative equals adding the positive).3. Subtract the constant terms: \(-30 - (-40) = 10\). The result of the subtraction is \(6p^2 + 4p + 10\).

Step 4: Simplify the result

Combine the results from the previous step if needed. Here, the expression \(6p^2 + 4p + 10\) is already in its simplest form with each term separated clearly.

Key Concepts

Vertical Form

Using vertical form for polynomial subtraction involves arranging the expressions much like how you would subtract multi-digit numbers. You begin by writing the polynomials one above the other so that each term is properly aligned by its degree. This means that terms involving the same power of a variable should be one above the other, with a clear column structure that makes it easier to perform the subtraction.

To ensure clarity, it's essential to remember every degree of your polynomial, even if it means using a zero as a placeholder when a specific term is not present. For example, in the exercise provided, aligning the polynomials like this:

• Align the highest degree terms, such as the squares, under one another.
• Place the linear terms together, even if it involves using 0 as a placeholder for one of the polynomials.
• And finally, align the constants directly below each other.

By setting the polynomials vertically with like terms matched up, it sets the stage for a quick and simple subtraction process.

Aligning Like Terms

Aligning like terms involves ensuring that similar orders of a variable are accurately placed in columns. This process acts as the infrastructure that supports a coherent subtraction of the polynomials.

When you align terms, make sure:

• The highest powers are directly on top of one another.
• The linear terms are also correctly placed, even when absent in a polynomial (use 0 as a placeholder).
• End with constant terms, which are numerical values without variables.

In our exercise, aligning the polynomials like terms allows you to see clearly which coefficients to subtract from one another. This alignment makes it practically impossible to get lost during the process, as each type of term is clearly marked and separated. It helps to think of each column as a mini-calculation of its own, where you tackle one straightforward subtraction before moving to the next.

Simplifying Polynomials

After successfully aligning and subtracting the like terms, the final step is simplifying the resulting expression. Simplification is crucial because it presents your answer in the most reduced and readable form.

For polynomial subtraction, simplifying involves combining like terms, but this might already be done during the subtraction phase if each column was computed correctly:

• For our specific problem, we've ended with the polynomial expression: \(6p^2 + 4p + 10\).
• Check if any terms can be further simplified or combined. In this case, we see it is already simplified because none of the terms are like terms.
• Ensuring all coefficients are simplified is key—no unnecessary factors should remain.

The work done previously in aligning terms and ensuring accurate subtraction now pays off because simplifying is quick with a polished and error-free result, leaving the answer clear and concise straight away.