Problem 62 - Algebra 1: Concepts and Skills

Question

Find the quotient. Divide \(\left(7 p^{5}+18 p^{4}\right)\) by \(p^{4}\)

Step-by-Step Solution

Answer

The quotient upon dividing \(7 p^{5}+18 p^{4}\) by \(p^{4}\) is \(7p + 18\).

Step 1: Understand what you are being asked

The student is being asked to divide the polynomial \(\left(7 p^{5}+18 p^{4}\right)\) by \(p^{4}\).

Step 2: Divide the first term

First, divide the first term of the polynomial \(7 p^{5}\) by \(p^{4}\). It will give the first part of the quotient, which is \(7p\). Because when the same base with exponents is divided, the powers are subtracted according to the law of exponents (that says \(a^{m}/a^{n} = a^{m-n}\)). In this case, it is \(7 p^{5}/p^{4} = 7 p^{5-4} =7p\).

Step 3: Divide the second term

Next, divide the second term of the polynomial \(18 p^{4}\) by \(p^{4}\). It will give the second part of the quotient, which is \(18\). The same law of exponents applies here. So, it is \(18 p^{4}/p^{4} = 18 p^{0}\) and according to the law of exponents anything powered to zero equates to one, therefore, it simplifies to \(18*1 = 18\).

Step 4: Write the final answer

Combine the quotients obtained in Step 2 and Step 3 to write down the final answer. So, the final quotient after dividing the given polynomial by \(p^{4}\) is \(7p + 18\).

Key Concepts

Law of Exponents

Understanding the \textbf{Law of Exponents} is crucial when working with algebraic expressions, especially in operations such as polynomial division. The law states that when you divide two expressions with the same base, you simply subtract the exponents: \( a^m / a^n = a^{m-n} \). For instance, in the exercise presented, we applied this rule to calculate \( 7p^5 / p^4 \) which results in \( 7p^{5-4} = 7p \). This simplification makes polynomial division more manageable.

It's also important to remember other exponent rules, such as any number (except zero) raised to the power of zero equals one \( b^0 = 1 \), which was used in dividing \( 18p^4 \) by \( p^4 \) resulting in 18. Keeping these laws in mind not only assists in the division process but also helps prevent common mistakes in manipulating algebraic expressions.

Dividing Polynomials

When \textbf{Dividing Polynomials}, like in our example \( (7p^5 + 18p^4) / p^4 \), the process resembles long division with numbers. Each term in the numerator (the polynomial being divided) is divided by the denominator (the divisor).

Here's a critical point: each term in the polynomial is divided separately. In our exercise, the term \( 7p^5 \) is divided by \( p^4 \) and then \( 18p^4 \) is divided by \( p^4 \). As you carry out these operations, it's essential to apply the law of exponents accurately for each term to simplify the polynomial correctly. The end result will combine the individual quotients from each division to form the final simplified expression. It's beneficial to approach each term methodically to avoid errors and to clearly see the final algebraic structure emerge.

Algebraic Expressions

Working with \textbf{Algebraic Expressions} requires recognizing patterns, understanding properties of numbers and operations, and correctly applying them. These expressions, comprising numbers, variables, and arithmetic operations, can take on different forms and complexities.

In the context of the exercise, we worked with an algebraic expression in the numerator that represented a polynomial. Polynomials are algebraic expressions that contain variables raised to whole number exponents, along with coefficients. The process of dividing the expression by another expression (like \( p^4 \) in this case) simplifies the complex expression into one that is more manageable and easier to interpret. Grasping the interplay of operations and their order, as well as being aware of special rules like the law of exponents, equips one with the necessary tools to proficiently manipulate these expressions for various mathematical applications.