Problem 40 - Algebra for College Students

Question

Subtract the polynomials using the vertical format. \(2 x^{2}-7 x-10\) from \(-x^{3}-12\)

Step-by-Step Solution

Answer

Result: \(-x^3 - 2x^2 + 7x - 2\)

Step 1: Write the Polynomials Vertically

To subtract the polynomials using the vertical format, first write them one above the other, aligning like terms. Place the polynomial to be subtracted, \(2x^2 - 7x - 10\), below the other polynomial, \(-x^3 - 12\):\[\begin{align*} & -x^3 + 0x^2 + 0x - 12 \- & (2x^2 - 7x - 10)\end{align*}\]

Step 2: Subtract the Polynomials

Now, perform the subtraction. Change the signs of the polynomial being subtracted:\[\begin{align*}-x^3 + 0x^2 + 0x - 12 \-2x^2 + 7x + 10 \ \ \hline-x^3 - 2x^2 + 7x - 2\end{align*}\]Subtract each column: - \(-x^3\) minus nothing remains \(-x^3\).- \(0x^2 - 2x^2 = -2x^2\).- \(0x + 7x = 7x\).- \(-12 + 10 = -2\).

Step 3: Verify and Simplify If Needed

Review the subtraction for accuracy. The result is:\[-x^3 - 2x^2 + 7x - 2\]There are no like terms to combine, so the polynomial is in its simplest form.

Key Concepts

Vertical Format

When subtracting polynomials, using the vertical format can make the process much simpler and more organized. This approach involves writing the polynomials you are working with in a column, much like how you would set up a traditional subtraction problem in arithmetic.

First, write each polynomial such that the corresponding terms are aligned in a straight vertical line. Generally, you will place the polynomial you are subtracting beneath the other polynomial.

• Ensure that each term is aligned by its degree. This means that all the terms with the same power of x are under each other.
• If there are any missing terms, fill them in with a zero coefficient to maintain alignment.

For example, in our exercise we had two polynomials: - Top Polynomial: \(-x^3 + 0x^2 + 0x - 12\) - Bottom Polynomial: \(2x^2 - 7x - 10\)
Aligning them vertically allows for an organized subtraction, making it easier to focus on individual terms one at a time.

Aligning Like Terms

Aligning like terms is crucial when dealing with polynomial subtraction. Polynomial terms are considered 'like' if they have the same variable raised to the same power. By aligning these terms vertically, subtraction becomes more straightforward and avoids confusion.

Here are some tips to help when aligning like terms:

• Ensure each column represents terms with the same power. For example, put all the x² terms together, all the x terms together, and so on.
• If a term is missing from one of the polynomials, make sure to fill in that space with a zero term, like \(0x^2\), to keep everything aligned.
• Double-check that no terms are left unaligned, as this could lead to errors in subtraction.

In our exercise, for instance, the x³ term only appears in the first polynomial. It's important to note that it's still correctly aligned with a zero placeholder in the second polynomial to ensure every column is complete.

Simplifying Polynomials

Once you have aligned the polynomials vertically and subtracted the terms, it's important to simplify the result. Simplification involves combining like terms if possible, but in subtraction, you'll often end up with a result that is already in its simplest form.

The outcome of our exercise, \(-x^3 - 2x^2 + 7x - 2\), showcases simplification. Each term stands alone with no like terms to combine:

• The \(-x^3\) term remains unchanged as there's no other x³ term.
• You subtract \(2x^2\) from the zero placeholder, resulting in \(-2x^2\).
• The x terms combine to give \(7x\).
• The constant terms, \(-12\) and \(10\), simplify to \(-2\).

Double-checking your subtraction for accuracy ensures that the polynomial is simplified and correct. Simplification is crucial as it leads to a neat and concise polynomial, making further operations or evaluations simpler.