Problem 15 - Algebra 2

Question

Simplify. $$ \left(28 c^{3} d-42 c d^{2}+56 c d^{3}\right) \div(14 c d) $$

Step-by-Step Solution

Answer

The simplified expression is \(2c^2 - 3d + 4d^2\).

Step 1: Identify the Expression to Simplify

The expression given is \( (28c^3d - 42cd^2 + 56cd^3) \div (14cd) \). We need to simplify this expression by dividing each term inside the parentheses by the term \( 14cd \).

Step 2: Divide Each Term Separately

First, divide \( 28c^3d \) by \( 14cd \):\[ \frac{28c^3d}{14cd} = 2c^2 \]Next, divide \( -42cd^2 \) by \( 14cd \):\[ \frac{-42cd^2}{14cd} = -3d \]Finally, divide \( 56cd^3 \) by \( 14cd \):\[ \frac{56cd^3}{14cd} = 4d^2 \].

Step 3: Write the Simplified Expression

Combine the results from dividing each term by \( 14cd \). The simplified expression is:\[ 2c^2 - 3d + 4d^2 \].

Key Concepts

Polynomial Division

Polynomial division is a method used to simplify expressions where a polynomial, which is a mathematical expression consisting of variables and coefficients, is divided by another polynomial. In simpler terms, it involves breaking down a complex polynomial expression to make it more manageable.

To divide polynomials, we follow a method similar to long division. In this exercise, the polynomial in the numerator is \( 28c^3d - 42cd^2 + 56cd^3 \), and the divisor is \( 14cd \). Each term in the numerator is individually divided by the divisor. This approach keeps the division process literal and organized.

• Start by dividing the coefficients (numbers in front of the variables). For example, \( \frac{28}{14} = 2 \).
• Subtract the exponents of like variables. For instance, \( c^3 \div c = c^{3-1} = c^2 \).

The key is in patiently dividing each separate term and combining the results to find the simplified expression. This way, polynomial division becomes consistent and predictable.

Factoring Expressions

Factoring is a critical skill in algebra that involves breaking down an expression into its component factors. These factors can be numbers, variables, or a combination of both that are multiplied together to get the original expression.

In simplifying polynomial expressions through division, understanding factoring is vital. Before division, you might need to factor either the numerator or the denominator, or both, to make the division feasible or more straightforward.

• Identify common factors shared by terms in the expression. In this exercise, we notice all terms have a common factor of \( 14cd \).
• Then, factor out this common factor from the entire expression before proceeding with the division.

Factoring simplifies the process of polynomial division by reducing the complexity of the polynomial. It allows us to focus on smaller, easier-to-manage pieces of the polynomial, resulting in a cleaner, more streamlined division process.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying these expressions involves factoring and reducing them to their simplest forms, similar to simplifying numerical fractions.

For the given problem, the expression \( \left(28 c^{3} d-42 c d^{2}+56 c d^{3}\right) \div(14 c d) \) can be seen as a rational expression where both the top and the bottom are polynomials. The simplification requires dividing each term in the numerator by the polynomial in the denominator.

• Similar to numerical fractions, cancel out common factors between the numerator and the denominator. This is achieved through careful division or factoring.
• Simplify each resulting term individually, as done in polynomial division.

Fully simplified rational expressions are easier to manipulate, compare, and utilize in further mathematical problems, making rational expressions a fundamental component of algebraic skills.