Problem 63 - Elementary and Intermediate Algebra

Question

Use vertical form to subtract the polynomials. $$ \begin{array}{l} \quad {0.8 x^{3} \quad \quad \quad\quad-2.3 x+0.6} \\ {-\left(0.2 x^{3}-1.2 x^{2}-3.6 x+0.9\right)} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result is \(0.6x^3 - 1.2x^2 + 1.3x - 0.3\).

Step 1: Align the Polynomials

Align the polynomials vertically by similar terms. Line up the terms with the same power of \(x\) directly over each other.\[\begin{array}{rl}& 0.8x^3 + 0x^2 - 2.3x + 0.6 \-& (0.2x^3 - 1.2x^2 - 3.6x + 0.9)\end{array}\]

Step 2: Distribute the Negative Sign

Distribute the negative sign across the second polynomial. This changes the signs of each term in that polynomial.\[\begin{array}{rl}0.8x^3 + 0x^2 - 2.3x + 0.6 & \- 0.2x^3 + 1.2x^2 + 3.6x - 0.9 &\end{array}\]

Step 3: Subtract the Coefficients

Subtract the coefficients of like terms. Perform the operations for each column.- For \(x^3\) terms: \(0.8 - 0.2 = 0.6\).- For \(x^2\) terms: \(0 - 1.2 = -1.2\).- For \(x\) terms: \(-2.3 + 3.6 = 1.3\).- For the constant terms: \(0.6 - 0.9 = -0.3\).

Step 4: Write the Resultant Polynomial

Combine the results from each column to write the final polynomial.The polynomial resulting from the subtraction is:\[0.6x^3 - 1.2x^2 + 1.3x - 0.3\]

Key Concepts

Vertical Alignment

When subtracting polynomials, we first need to align them vertically by matching the terms with identical powers of \(x\). This setup ensures that we only perform operations on like terms:

* Example of aligning polynomials vertically:
- Place the polynomial \(0.8x^3 + 0x^2 - 2.3x + 0.6\) above
- Place \(0.2x^3 - 1.2x^2 - 3.6x + 0.9\) below, making sure to line up the terms: * Notice that
- \(x^3\) terms are aligned under \(x^3\) terms
- \(x^2\) with \(x^2\)
- \(x\) with \(x\)
- Constants with constants.

This step may sometimes require inserting zero terms, such as \( 0x^2 \), to fill in missing terms and maintain the alignment.

Distribute Negative Sign

Once the polynomials are correctly aligned, the next crucial step involves distributing the negative sign from the subtraction operation. You need to alter the signs of every term in the polynomial being subtracted.

* Effect of distributing the negative sign:
- Each term of the second polynomial changes its sign:
- The positive becomes negative and vice versa. * For example:
- From \(0.2x^3 - 1.2x^2 - 3.6x + 0.9\) becomes \(-0.2x^3 + 1.2x^2 + 3.6x - 0.9\).

This step effectively changes your subtraction problem into an addition problem, where you will add the negated polynomial.

Subtract Coefficients

After distributing the negative sign to the second polynomial, you are ready to subtract the coefficients of like terms. This action is akin to performing the regular arithmetic operation on the numbers directly over each other.

* Let's perform subtraction for each column:
- For \(x^3\) terms: calculate \(0.8 - 0.2\) to get \(0.6\).
- For \(x^2\) terms: compute \(0 - 1.2\) which results in \(-1.2\).
- For \(x\) terms: evaluate \(-2.3 + 3.6\) to find \(1.3\).
- Finally, for constants, tackle \(0.6 - 0.9\) to reach \(-0.3\).

This careful subtraction of coefficients ensures that each term of the polynomial is accurately calculated.

Polynomial Operations

Polynomial operations, including addition and subtraction, are essential concepts when dealing with algebraic expressions consisting of multiple terms. Understanding the process of polynomial subtraction is crucial as it encompasses aligning the terms, distributing any negative signs, and subtracting coefficients.

* Basics of polynomial subtraction:
- Involves organizing like terms and applying operations correctly.
- Key operations include distributing negations and calculating differences for each set of terms.

As you're aware, executing polynomial operations skillfully paves the way for solving more complex algebraic equations and even learning about polynomial factoring and division in advanced mathematics.