Problem 31 - Elementary and Intermediate Algebra

Question

Divide the polynomial by the monomial. See Example 2. $$ \frac{9 s^{8}-18 s^{5}+12 s^{4}}{3 s^{3}} $$

Step-by-Step Solution

Answer

\( 3s^5 - 6s^2 + 4s \)

Step 1: Break Down the Division

The given expression is \( \frac{9s^8 - 18s^5 + 12s^4}{3s^3} \). To cleanly divide each term of the polynomial by the monomial, we'll distribute \( \frac{1}{3s^3} \) to each term separately: \( \frac{9s^8}{3s^3} - \frac{18s^5}{3s^3} + \frac{12s^4}{3s^3} \).

Step 2: Divide Each Term

Starting with the first term: \( \frac{9s^8}{3s^3} = 3s^5 \) because \( 9 \div 3 = 3 \) and \( s^8 \div s^3 = s^{8-3} = s^5 \).

Step 3: Simplify the Second Term

For the second term: \( \frac{-18s^5}{3s^3} = -6s^2 \). The division of coefficients gives \(-18 \div 3 = -6\) and the division of the powers of \( s \) results in \( s^{5-3} = s^2 \).

Step 4: Simplify the Third Term

For the last term: \( \frac{12s^4}{3s^3} = 4s \). The coefficient division is \( 12 \div 3 = 4 \). With the powers of \( s \), we have \( s^{4-3} = s^1 = s \).

Step 5: Combine the Simplified Terms

Now, combine all the simplified terms together, resulting in: \( 3s^5 - 6s^2 + 4s \). This is the final simplified form of the given expression.

Key Concepts

Polynomial Division

Polynomial division is an essential technique in algebra for dividing one polynomial by another. In the given exercise, we are dividing a polynomial by a monomial, which is a polynomial with just one term.
This type of division is generally straightforward, as each term in the dividend is divided separately by the divisor.
Here are some key points to consider:

• When dividing a polynomial by a monomial, divide each term of the polynomial by the monomial individually.
• Make sure to handle both the numerical coefficients and the variables separately. This means performing basic division on the numbers and subtracting the exponents of like bases.

Polynomial division may seem daunting at first, but understanding its components can make it manageable. Practicing with simpler polynomials and gradually increasing complexity can help build confidence in applying this concept.

Monomials

Monomials are the simplest form of polynomials, consisting of a single term that can be a constant, a variable, or the product of constants and variables raised to whole number powers.
In our example, the monomial is the denominator, \(3s^3\).
Recognizing and understanding monomials is key for simplifying polynomial expressions.

• To divide by a monomial, every term in the polynomial is divided by the whole monomial. Each term in the polynomial must interact with both the number and variable part of the monomial.
• A monomial is easy to manipulate due to its singular term structure, making it a helpful tool when simplifying larger polynomial expressions.

Once you have a handle on managing monomials, you will find it much simpler to progress to complex polynomial manipulation.

Simplifying Expressions

Simplifying expressions is an important skill in algebra that involves reducing expressions to their simplest form. This often requires dividing, factoring, and combining like terms.
In our current example, each individual division results in a simpler term, which we can combine to find the overall simplified expression.

• Simplification uses the arithmetic of division for coefficients along with the rules of exponents to handle the variables.
• Subtracting the exponent of the divisor from the exponent of the dividend is a key step when simplifying powers.

The final result \(3s^5 - 6s^2 + 4s\) showcases how significantly the original expression can be simplified through systematic division and reduction techniques.
Consistent practice with these steps results in efficiency and accuracy when manipulating algebraic expressions.