Problem 49 - Basic Engineering Mathematics

Question

Simplify \(\frac{1}{3}\) of \(3 p+4 p(3 p-p)\)

Step-by-Step Solution

Answer

\(\frac{8}{3}p^2 + p\)

Step 1: Expand the Expression

First, expand the expression inside the parentheses. We have \(3p + 4p(3p - p)\). Start by expanding \(4p(3p - p)\).First distribute \(4p\) into \(3p - p\):\[4p(3p - p) = 4p imes 3p - 4p imes p = 12p^2 - 4p^2\].So the expanded expression becomes:\[3p + 12p^2 - 4p^2\].

Step 2: Simplify the Expanded Terms

Now we will simplify the terms \(12p^2 - 4p^2\) in the expression. Combine like terms:\[12p^2 - 4p^2 = 8p^2\].The expression is simplified to:\[8p^2 + 3p\].

Step 3: Distribute \\(\frac{1}{3}\\)

Now, compute \(\frac{1}{3}\) of our simplified expression. This means multiplying each term by \(\frac{1}{3}\):\[\frac{1}{3}(8p^2 + 3p) = \frac{1}{3} \times 8p^2 + \frac{1}{3} \times 3p\].Simplifying these gives:\[\frac{8}{3}p^2 + p\].

Step 4: Final Simplified Expression

The expression is now fully simplified. The terms \(\frac{8}{3}p^2\) and \(p\) are as simple as they can be. The final answer is:\[\frac{8}{3}p^2 + p\].

Key Concepts

Polynomial Expansion

Polynomial expansion is the process of removing parentheses in an expression by multiplying the terms inside the parentheses by the term outside.
When you have an expression like \(4p(3p - p)\), it means you need to multiply each term inside the parentheses by \(4p\). Here’s how it works:

• Multiply \(4p\) by \(3p\) to get \(12p^2\).
• Multiply \(4p\) by \(p\) to get \( - 4p^2\).

The expanded polynomial would then be \(12p^2 - 4p^2\). The key is to be meticulous with signs and coefficients to ensure accuracy.
Expansion lays the groundwork for further simplification.

Distributive Property

The distributive property is essential for expanding expressions and it's used to distribute a single term across terms within parentheses.
The general form of the distributive property is \(a(b + c) = ab + ac\).
In our exercise, this property was first used when \(4p\) was distributed over \(3p - p\) to yield \(12p^2 - 4p^2\).

• Each term is multiplied separately and then added or subtracted.
• It helps break down expressions, making them easier to handle.

Moreover, later in the exercise, the property is applied again to find \(\frac{1}{3}\) of the simplified expression. Using the distributive property efficiently enables the elimination of parentheses and the simplification of complex expressions.

Combining Like Terms

Combining like terms is a critical step in simplifying algebraic expressions by reducing them to their simplest form.
Terms with the same variable and exponent can be combined, by simply adding or subtracting their coefficients.
In our problem, after expanding, we identified like terms from \(12p^2\) and \(-4p^2\).

• The result of combining these terms is \(8p^2\).
• It reduces the number of terms, making the expression easier to handle.

The process of combining like terms is foundational in algebra and helps to keep expressions manageable and understandable.

Expression Simplification

Expression simplification is the process of making an algebraic expression as concise and clear as possible.
Following polynomial expansion, using the distributive property, and combining like terms, expression simplification involves condensing the expression into its simplest form without losing its value.

• Simplify fractions and coefficients, like multiplying the whole expression by \(\frac{1}{3}\) in our example.
• Ensure each step keeps the expression equivalent to the original.

Our given expression \(\frac{1}{3}(8p^2 + 3p)\) simplifies to \(\frac{8}{3}p^2 + p\). Each method used aligns the terms in a simpler structure, pointing towards the importance of simplification in solving algebraic expressions efficiently.