Problem 59 - Elementary and Intermediate Algebra

Question

Use vertical form to subtract the polynomials. $$ \text { Subtract }\left(s^{2}+4 s+2\right) \text { from }\left(5 s^{2}+9\right) $$

Step-by-Step Solution

Answer

The result is \( 4s^2 - 4s + 7 \).

Step 1: Rewrite the Expression

To subtract polynomials, first rewrite them. Align terms of like powers vertically: \[\begin{array}{r}5s^{2} + 0s + 9\-(s^{2} + 4s + 2)\end{array}\]

Step 2: Subtract the Terms

Subtract the polynomials by subtracting the coefficients of terms aligned vertically:\[\begin{array}{r}5s^2 + 0s + 9\-(s^2 + 4s + 2)\\hline(5s^2 - s^2) + (0s - 4s) + (9 - 2)\end{array}\]Now calculate each subtraction:- \(5s^2 - s^2 = 4s^2\)- \(0s - 4s = -4s\)- \(9 - 2 = 7\)

Step 3: Write the Resulting Polynomial

Combine the results from the subtraction of each aligned term:\[4s^2 - 4s + 7\]This is the resulting polynomial after subtraction.

Key Concepts

Vertical Form Subtraction

When dealing with polynomial subtraction, organizing the terms in vertical form makes the process easier and more structured.

Consider polynomials similar to numbers aligned in a column: each place beneath another of the same type. Here, it’s crucial to align terms based on their degree.

• Write the terms of the polynomial with the higher degree on top.
• Directly underneath, list the terms of the polynomial you are subtracting. Make sure the variable powers line up neatly.
• Place zeroes in place if a specific degree term is missing.

This organization emphasizes clarity and promotes accuracy, reducing errors during subtraction. Remember, each column represents a different degree of the variable, just like columns in regular arithmetic.So, for the exercise you were given, align the terms like this: \[\begin{array}{r}5s^{2} + 0s + 9\-(s^{2} + 4s + 2)\end{array}\] This setup makes it straightforward to subtract corresponding coefficients, simplifying the overall subtraction process.

Subtraction of Polynomials

Subtraction of polynomials involves a simple yet sometimes tricky process of subtracting each coefficient of like terms between two expressions.

To perform subtraction:

• Identify and align each polynomial's corresponding terms by their degree.
• Subtract the coefficients of each pair of like terms.
• Bear in mind that we treat the subtraction of each term as a mini subtraction problem.

For instance, the subtraction of polynomials presented: \[\begin{array}{r}5s^2 + 0s + 9\-(s^2 + 4s + 2)\end{array}\]leads to calculating:

• Subtracting the coefficients of the squared terms: \(5 - 1 = 4\)
• Subtracting the coefficients of the \(s\) terms: \(0 - 4 = -4\)
• Subtracting the constant terms: \(9 - 2 = 7\)

Each result becomes part of the new polynomial: \(4s^2 - 4s + 7\). This simple step-by-step process can be replicated for any polynomial subtraction task.

Algebraic Expressions

Understanding algebraic expressions is key to mastering polynomials.

An algebraic expression combines numbers, variables, and operational signs (such as + and -) to represent mathematical relations. Polynomials are a specific type of algebraic expression where the operations are addition, subtraction, and multiplication by a constant or a variable raised to a non-negative integer power. Each part of a polynomial is termed a "term." These terms are composed of:

• A coefficient - a constant multiplier (e.g., the number 3 in \(3s^2\)).
• A variable - a symbol, often \(s\), that can have different values.
• An exponent - a number indicating the power to which the variable is raised (e.g., the 2 in \(s^2\)).

Recognizing and understanding these components helps in performing operations like addition and subtraction. For example, in the expression \(5s^2 + 0s + 9\), \(5\) is the coefficient of \(s^2\), \(0\) is the coefficient of \(s\) (including it highlights the absence of a linear term), and \(9\) is the constant term. Mastering these basics ensures that tasks like polynomial subtraction and addition are approached with clarity and confidence.