Problem 22 - Algebra 1

Question

Divide. Divide \(16 y^{3}-36 y^{2}-64\) by \(-4 y^{2}\).

Step-by-Step Solution

Answer

The result of the given division is \(-4 y + 9 + 16/y^{2}\)

Step 1: Isolate Each Term

Divide the polynomial \(16 y^{3}-36 y^{2}-64\) to isolate each term. Then we have three terms: \(16 y^{3}\), \(-36 y^{2}\), and \(-64\).

Step 2: Dividing First Term

Divide the first term \(16 y^{3}\) by \(-4 y^{2}\). The y values are subtracted from each other (using the rules of exponents). This results in \(-4 y\).

Step 3: Dividing Second Term

Next, divide the second term \(-36 y^{2}\) by \(-4 y^{2}\). Because the y values match, they cancel each other out, resulting in 9.

Step 4: Dividing Third Term

Finally, divide the third term \(-64\) by \(-4 y^{2}\), which gives us \(16/y^{2}\).

Step 5: Combine All Terms

Combine all the terms obtained from the division to give the final answer. The resulting expression is: \(-4 y + 9 + 16/y^{2}\)

Key Concepts

Exponents

Exponents are a fundamental part of algebra. They represent a way to express repeated multiplication of the same number. For example, in the expression \( y^3 \), the exponent 3 tells us that \( y \) is multiplied by itself three times: \( y \times y \times y \).

When working with exponents, there are essential rules to remember. Here are a few key ones:

• Product of Powers Rule: To multiply two exponents with the same base, add their powers: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers Rule: To divide two exponents with the same base, subtract the power of the denominator from the power of the numerator: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).

In the given exercise, dividing by \(-4y^2\) involves manipulating exponents of \( y \). For instance, one of the steps simplifies \( 16y^3 \) divided by \(-4y^2\) to \(-4y\). We apply the quotient of powers rule here by subtracting the 2 in \( y^2 \) from the 3 in \( y^3 \), resulting in \( y^{3-2} = y^1 = y \). This basic exponent rule streamlines polynomial division.

Rational Expressions

A rational expression in algebra is similar to a fraction but instead of just numbers, it contains polynomials in the numerator and/or denominator. For example, \( \frac{x^2 + 1}{x - 1} \) is a rational expression.

To work with rational expressions effectively, you'll frequently perform operations like addition, subtraction, multiplication, and division. Here are some key points regarding rational expressions:

• Always begin by factoring both the numerator and the denominator if possible.
• Check for any common factors that can be canceled out, which simplifies the expression.
• Be cautious about restrictions in the expression, such as values for which the denominator becomes zero (these values are not part of the domain).

In the exercise, dividing the last term \(-64\) by \(-4y^2\) introduces the rational expression \( \frac{16}{y^2} \). This term is part of the overall expression after each polynomial term has been divided by \(-4y^2\). Dealing with rational expressions like these requires careful manipulation of terms to ensure accuracy. For example, understanding that you can't have a zero denominator helps in grasping valid operations on rational expressions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This process often includes combining like terms and applying arithmetic operations and algebraic rules.

Here’s a general approach to simplifying expressions:

• Identify Like Terms: Like terms have the same variable raised to the same power, such as \(2x^2\) and \(-5x^2\). Combine them by performing the arithmetic operations.
• Apply Distribution: Use the distributive property \( a(b + c) = ab + ac \) to simplify expressions, especially in terms of multiplication.
• Factor and Cancel: Factor expressions to identify and cancel out common terms across the numerator and the denominator in rational expressions, if applicable.

In the given exercise, the final expression derived from dividing each term of the polynomial by \(-4y^2\) is simplified to \(-4y + 9 + \frac{16}{y^2} \). Here, the steps taken ensure that the terms in the expression are as simplified as possible by following proper algebraic techniques. Understanding how to combine terms correctly and transform a complex expression into its simplest form is a critical skill in algebra.