Problem 60 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ s^{10}\left(2 s^{5}+3 s^{4}+4 s^{3}+5 s^{2}+2 s+2\right)-s^{15}+2 s^{14}+3 s\left(s^{12}+4 s^{11}\right)-s^{10} $$

Step-by-Step Solution

Answer

Answer: The simplified algebraic expression is $s^{15} + 5s^{14} + 7s^{13} + 17s^{12} + 2s^{11} + 2s^{10}$.

Step 1: Combine Like Terms

Notice that there are terms inside parentheses that can be simplified. We can distribute the term s^{10} across the terms inside the parentheses. After doing so, the expression will look as follows: $$ 2s^{15} + 3s^{14} + 4s^{13} + 5s^{12} + 2s^{11} + 2s^{10} - s^{15} + 2s^{14} + 3s^{13} + 12s^{12} $$

Step 2: Combine Terms Again and Rearrange

Now, we can combine like terms and rearrange them in the descending order of the power of 's': $$ (2s^{15} - s^{15}) + (3s^{14} + 2s^{14}) + (4s^{13} + 3s^{13}) + (5s^{12} + 12s^{12}) + (2s^{11}) + (2s^{10}) $$

Step 3: Simplify

Finally, we can simplify the expression by performing the operations: $$ s^{15} + 5s^{14} + 7s^{13} + 17s^{12} + 2s^{11} + 2s^{10} $$ So, the simplified algebraic expression for the given problem is: $$ s^{15} + 5s^{14} + 7s^{13} + 17s^{12} + 2s^{11} + 2s^{10} $$

Key Concepts

Understanding Polynomials

Polynomials are expressions with one or more terms combined using addition, subtraction, multiplication, but never division by a variable. Each term in a polynomial consists of a coefficient (a constant number) and a variable raised to a whole number exponent. For example, in the polynomial expression \(3s^4\), \(3\) is the coefficient, \(s\) is the variable, and \(4\) is the exponent.
Polynomials can have multiple terms like:

• Monomials: Single-term polynomials like \(5s^2\).
• Binomials: Two-term polynomials like \(2s + 3\).
• Trinomials: Three-term polynomials like \(s^2 + 2s + 1\).
• General Polynomials: Four or more terms like \(s^4 + 3s^3 + 2s^2 + s + 1\).

Understanding polynomial structure helps in simplifying and manipulating these expressions in algebra.

Using Distributive Property

The distributive property is a useful tool for simplifying expressions. It involves multiplying a single term by each term inside a parenthesis. For example, when you have an expression like \(s^{10}(2s^5 + 3s^4)\), you apply the distributive property as follows:

• Multiply \(s^{10}\) by \(2s^5\) to get \(2s^{15}\).
• Then, multiply \(s^{10}\) by \(3s^4\) to get \(3s^{14}\).

Applying the distributive property correctly rearranges and simplifies complex polynomial expressions, making it easier to handle before combining like terms. This process ensures each component within the parenthesis is correctly addressed, resulting in a more streamlined expression for further simplification steps.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are those that have the same variable raised to the same power. You can simplify these terms by adding or subtracting their coefficients.
For instance, if you have \(3s^{13}\) and \(4s^{13}\), both have the same variable, \(s\), and the exponent, \(13\). Therefore, they are like terms. You can combine them as follows:

• Add their coefficients: \(3 + 4 = 7\).
• The result is \(7s^{13}\).

Combining like terms reduces the complexity of the expression and is often a final step in simplifying the expression after using the distributive property. It ensures that the polynomial is expressed in the simplest possible form by consolidating similar terms.

Working with Exponents

Exponents denote how many times a number or variable is multiplied by itself. In algebra, understanding how to manipulate exponents is crucial for simplifying expressions. Here's a quick recap of key properties:

• Multiplying with the same base: Add the exponents. For example, \(s^a \cdot s^b = s^{a+b}\).
• Dividing with the same base: Subtract the exponents. Example: \(\frac{s^a}{s^b} = s^{a-b}\).
• Power of a power: Multiply the exponents. Example: \((s^a)^b = s^{ab}\).

In the context of simplifying polynomials, managing exponents accurately allows terms to be combined effectively when using the distributive property or when combining like terms. Mistakes with exponents can lead to incorrect results, so it's important always to handle them with care during algebraic simplifications.