Problem 43 - Elementary Algebra

Question

\((p-4)\left(p^{2}+5 p-1\right)\)

Step-by-Step Solution

Answer

The expanded form is \(p^3 + p^2 - 21p + 4\).

Step 1: Distribute the First Term

Distribute the term \(p-4\) to each term inside the parentheses \(p^2+5p-1\). This means you multiply \(p-4\) by each of the terms \(p^2\), \(5p\), and \(-1\) separately.

Step 2: Multiply \(p\) by Each Term

First, multiply \(p\) by each term inside the parentheses: \(p \times p^2 + p \times 5p + p \times (-1)\). This yields \[p^3 + 5p^2 - p\].

Step 3: Multiply \(-4\) by Each Term

Next, multiply \(-4\) by each term inside the parentheses: \(-4 \times p^2 + (-4) \times 5p + (-4) \times (-1)\). This yields \[-4p^2 - 20p + 4\].

Step 4: Combine Like Terms

Finally, combine the terms from Step 2 and Step 3: \[p^3 + 5p^2 - p - 4p^2 - 20p + 4\]. Simplify this by combining like terms: \[p^3 + (5p^2 - 4p^2) + (-p - 20p) + 4\], resulting in \[p^3 + p^2 - 21p + 4\].

Key Concepts

distribution

In algebra, distribution is a powerful and essential tool for simplifying expressions. It involves multiplying each term within a parenthesis by a term outside the parenthesis. This method helps in spreading the terms for easier handling.

Consider the expression \( (p-4)(p^2 + 5p - 1) \). Here, we distribute the term \( p-4 \) to each term inside the parentheses \( p^2 + 5p - 1 \).

To visualize, think about distributing cakes among kids. If you have three slices (terms) of cakes and two kids (expanders) for each slice, you give a slice to each, and in the end, each kid will have their collection of slices as individual portions.

So, for our polynomial, we split \( p-4 \) and distribute this to each term inside \( p^2 + 5p - 1 \):

• First, distribute \( p \) to \( p^2, 5p, \) and \(-1\); \( p \times p^2, p \times 5p, p \times (-1) \)
• Then distribute \(- 4 \) to the same terms; \(-4 \times p^2, -4 \times 5p, -4 \times (-1) \)

The next step is to perform each multiplication individually and then move to the next key concept.

combining like terms

After distribution, the next step is to organize and simplify what we have. This brings us to the process of combining like terms.

Like terms are terms that have the same variables raised to the same power. In our example, after distributing, we get:

\[ p^3 + 5p^2 - p - 4p^2 - 20p + 4 \]

We arrange these terms so terms with similar powers come together. Here, \( p^3 \), \( 5p^2 \), and \(- 4p^2\) are terms with the same variable raised to the same power:

\[ p^3 + 5p^2 - 4p^2 - p - 20p + 4 \]

Combine them to simplify the expression. Group like terms:

• \(5p^2 - 4p^2 = p^2\)
• \( -p - 20p = -21p\)
• The constants are standalone numbers. In our example, there's only \( +4 \)

Simplifying, our expression looks like this:

\[ p^3 + p^2 - 21p + 4 \]

Combining like terms ensures we have the simplest and most manageable form of our polynomial expression.

step-by-step algebra

Learning algebra step-by-step helps in understanding the logical flow of solving expressions and equations.

Here's the breakdown of solving \( (p-4)(p^2 + 5p - 1) \):

Step 1: Distribute
First, distribute \( p \) and \(-4\) to each term inside the parentheses.

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