Problem 38 - Algebra for College Students

Question

Subtract the polynomials using the vertical format. \(2 x^{3}-x+6\) from \(x^{3}+4 x^{2}+1\)

Step-by-Step Solution

Answer

The result is \(-x^3 + 4x^2 + x - 5\).

Step 1: Align the Polynomials

Write the polynomials in a vertical format, lining up terms with the same degree:\[\begin{array}{c} x^3 + 4x^2 + 0x + 1 \-(2x^3 + 0x^2 - x + 6)\end{array}\]Note: Consider all degrees, including missing ones like \(0x^2\) and \(0x\).

Step 2: Subtract the Cubic Terms

Subtract the cubic terms:\[x^3 - 2x^3 = -x^3\]

Step 3: Subtract the Quadratic Terms

Subtract the quadratic terms:Since the first polynomial has \(4x^2\) and the second polynomial has \(0x^2\), we have:\[4x^2 - 0 = 4x^2\]

Step 4: Subtract the Linear Terms

Subtract the linear terms:The first polynomial provides \(0x\) and the second has \(-x\):\[0 - (-x) = x\]

Step 5: Subtract the Constant Terms

Subtract the constant terms:\[1 - 6 = -5\]

Step 6: Write the Final Result

Combine all the resulting terms to form the final result:\[-x^3 + 4x^2 + x - 5\]

Key Concepts

Vertical Format

Polynomial subtraction, much like regular subtraction, can be neatly organized using a vertical format. This method is particularly useful to ensure each corresponding term is correctly aligned. Imagine it similar to how you would write numbers directly on top of each other before subtracting. For polynomials, this means stacking each polynomial on top of the other, aligning terms with the same power (degree).
Using vertical format promotes accuracy by making sure terms do not mix up, especially when handling various degrees like the cubic, quadratic, and linear terms. This visual method ensures each degree is easy to recognize, tackle, and subtract from its counterpart.

Polynomial Alignment

When subtracting polynomials, proper alignment is key to ensure each term is subtracted from its respective counterpart. Let's delve deeper into this concept:

• Align terms with the same degree directly over one another.
• For terms that do not appear in both polynomials, use placeholders (like 0) to maintain correct positioning.

For instance, if one polynomial lacks a quadratic term, introduce it as "+0x^2". This way, even absent terms are acknowledged, keeping the operation clear and streamlined. Correct alignment prevents mistakes and simplifies the subtraction process.
By visually organizing terms, you avoid mismatches and facilitate straightforward arithmetic operations on matching terms.

Polynomial Terms

Understanding polynomial terms is crucial in solving polynomial subtraction problems. A polynomial, essentially, is a sum of terms, each consisting of a variable raised to a certain power coupled with a coefficient. Here's a quick breakdown:

• Cubic terms: These are terms like \(x^3\), representing the highest power of the polynomial.
• Quadratic terms: Denoted by \(x^2\), these are next in line in terms of degree.
• Linear terms: These involve just \(x\), making up the first-degree terms.
• Constant terms: Numbers without variables, such as 6 or 1 in this example. They account for the zero-degree terms.

Correctly identifying and organizing these is vital in the vertical subtraction process, ensuring each type of term is subtracted correctly from its counterpart.

Step-by-Step Solution

Breaking down polynomial subtraction step by step aids in traversing from confusion to clarity. Let’s embark on this journey:The process begins with aligning both polynomials vertically. By employing placeholders for missing terms, you smoothly align each degree.
1. **Cubic Terms:** Commence with the cubic terms. Subtract \(2x^3\) from \(x^3\) to attain \(-x^3\). The biggest terms take precedence here.
2. **Quadratic Terms:** Proceed to the quadratic terms. Subtract \(0x^2\) from \(4x^2\), resulting in an unaltered \(4x^2\). Even absent terms are part of this process.3. **Linear Terms:** Continue with linear terms. Here, subtract \(-x\) from \(0x\), resulting in \(x\).4. **Constant Terms:** Lastly, subtract constant terms: \(1-6 = -5\)..Combine these results to form \(-x^3 + 4x^2 + x -5\). With each step, you ensure consistent methodology, thereby arriving at a precise result.