Problem 18 - Elementary Algebra

Question

\(\left(4 p^{4}-16 p^{2}+60 p+20\right) \div(4 p)\)

Step-by-Step Solution

Answer

The simplified expression is \( p^{3} - 4 p + 15 + \frac{5}{p} \)

Step 1: Understand the Problem

The given expression \(\bigg(4 p^{4} - 16 p^{2} + 60 p + 20\bigg) \div (4 p)\) indicates a polynomial divided by a monomial. This means we need to divide each term of the polynomial by the monomial.

Step 2: Divide Each Term

Individually divide each term in the numerator by \( 4 p \). \(-\) \(\frac{4 p^{4}}{4 p} = p^{3}\) \(-\) \(\frac{16 p^{2}}{4 p} = 4 p\) \(-\) \(\frac{60 p}{4 p} = 15\) \(-\) \(\frac{20}{4 p} = \frac{5}{p}\)

Step 3: Combine the Results

Combine the results from step 2 to get the final simplified expression: \( p^{3} - 4 p + 15 + \frac{5}{p} \)

Key Concepts

Polynomials

Understanding polynomials is crucial in algebra. A polynomial is an algebraic expression that consists of variables, coefficients, and exponents. For example, in the given exercise, \(4p^4 - 16p^2 + 60p + 20\), the polynomial contains different terms like \(4p^4\), \(-16p^2\), \(60p\), and 20. Each term is composed of a coefficient (the number in front) and a variable raised to an exponent (like \(p^4\) or \(p^2\)). The degree of a polynomial is the highest exponent in the expression, which in this case is 4. Polynomials can be used to model a wide range of real-world situations, making them an essential concept to grasp.

Monomials

When we talk about monomials, we're referring to a simpler form of polynomials. A monomial is an algebraic expression with only one term. This term includes a coefficient, a variable, and an exponent. In the given exercise, the monomial is \(4p\). To recognize a monomial, look for a single term without any addition or subtraction signs. For example, \(7x^3\), \(-5y\), and \(12\) are all monomials. Notice that the monomial can have different variables and/or exponents, but it should always be a single term.

Algebraic Simplification

Algebraic simplification involves breaking down complex expressions into simpler, more manageable forms. This process includes performing operations like addition, subtraction, multiplication, and division across terms of an equation or expression.
In our exercise, we simplified by dividing each term of the polynomial \(4p^4 - 16p^2 + 60p + 20\) by the monomial \(4p\).
Simplify step by step:

• First, \(\frac{4p^4}{4p} = p^3\)
• Then, \(\frac{16p^2}{4p} = 4p\)
• After that, \(\frac{60p}{4p} = 15\)
• Finally, \(\frac{20}{4p} = \frac{5}{p}\)

Combining these simplified terms, we get the result: \(p^3 - 4p + 15 + \frac{5}{p}\). Breaking down each term makes it easier to manage and understand the problem.

Division of Terms

Division of terms in algebra is when you split each term of an algebraic expression by another term. In our example, we divided \(4p^4 - 16p^2 + 60p + 20\) by \(4p\). This means each term in the numerator (the polynomial) is divided by the monomial (in this case, \(4p\)).
Here’s how you can divide terms:

• Divide the coefficients (numbers in front of variables).
• Subtract the exponents of like variables (numerator minus denominator).

For example, when dividing \(4p^4\) by \(4p\), you divide the coefficients to get 1, and subtract the exponents of \(p\) which gives \(p^{4-1} = p^3\). Keep repeating this for each term: \(\frac{4p^4}{4p} = p^3\), \(\frac{16p^2}{4p} = 4p\), \(\frac{60p}{4p} = 15\), and \(\frac{20}{4p} = \frac{5}{p}\). This step-by-step approach helps avoid mistakes and ensures you get a simplified, correct result.