Problem 64 - Elementary and Intermediate Algebra

Question

Perform each division. $$ \frac{30 y^{8}+40 y^{7}}{10 y^{6}} $$

Step-by-Step Solution

Answer

The division results in \( 3y^2 + 4y \).

Step 1: Identify the terms in the numerator and the denominator

The numerator of the fraction is \(30y^8 + 40y^7\) and the denominator is \(10y^6\). We need to divide each term in the numerator by the term in the denominator.

Step 2: Divide the first term of the numerator by the denominator

Start by dividing the first term in the numerator, \(30y^8\), by the denominator, \(10y^6\). Perform the division of the coefficients and apply the subtraction of exponents rule. So, \( \frac{30y^8}{10y^6} = 3y^{8-6} = 3y^2 \).

Step 3: Divide the second term of the numerator by the denominator

Now, divide the second term in the numerator, \(40y^7\), by the denominator, \(10y^6\). Perform the division of the coefficients and apply the subtraction of exponents rule. So, \( \frac{40y^7}{10y^6} = 4y^{7-6} = 4y^1 = 4y \).

Step 4: Write the final simplified expression

Combine the results from Step 2 and Step 3 to obtain the simplified expression for the fraction. Thus, the division results in \( 3y^2 + 4y \).

Key Concepts

Polynomial Division

Polynomial division is much like standard numerical division. However, it involves algebraic expressions instead of plain numbers. To divide polynomials, we need to divide each term of the polynomial in the numerator by the polynomial in the denominator. In our specific problem:- The numerator is a polynomial: \(30y^8 + 40y^7\).- The denominator is a monomial: \(10y^6\).Our task is to "break down" the original polynomial into simpler parts by handling each term in the numerator independently. This simplification helps us easily manage more complex expressions by reducing them into less complicated forms. Ultimately, the division of these numerators by the shared denominator simplifies at each step, making it more digestible and manageable.

Exponent Rules

Exponents are a way of representing repeated multiplication. They follow specific rules that simplify complex operations such as division. One crucial rule when dividing like terms is the subtraction of exponents.- For the expression \( \frac{a^m}{a^n} \), the rule is to subtract the exponents, yielding \( a^{m-n} \).In the context of our problem:- For \( \frac{30y^8}{10y^6} \), subtract the exponent in the denominator from the exponent in the numerator: \( 8 - 6 = 2 \).
- Similarly, for \( \frac{40y^7}{10y^6} \), subtract \( 6 \) from \( 7 \), resulting in \( y^1 \), or simply \( y \).Exponents help simplify expressions and make calculations more efficient, especially when dealing with repeated variables.

Simplification of Expressions

Simplification involves rewriting an expression in its simplest form, reducing complexity while retaining value. When dealing with polynomials, this means reducing the expression to the lowest possible number of terms and simplest forms.- After dividing each term in our numerator by the denominator, we get \( 3y^2 \) and \( 4y \).
- Together, they are combined into a simpler expression: \( 3y^2 + 4y \).By systematically applying the rules of algebra and exponents, we've transformed a complex polynomial fraction into a more straightforward addition of terms. Simplifying expressions not only makes working with them easier but also aids in understanding underlying mathematical relationships.