Problem 7 - Algebra 2

Question

Simplify. $$ 2 p^{2} q\left(5 p q-3 p^{3} q^{2}+4 p q^{4}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(10p^3q^2 - 6p^5q^3 + 8p^3q^5\).

Step 1: Distribute 2p²q Across Each Term in the Parentheses

Start by multiplying the expression outside the parenthesis, which is \(2p^2q\), with each term inside the parenthesis individually.1. Multiply \(2p^2q\) by \(5pq\): \[ 2p^2q imes 5pq = 10p^3q^2 \]2. Multiply \(2p^2q\) by \(-3p^3q^2\): \[ 2p^2q imes (-3p^3q^2) = -6p^5q^3 \]3. Multiply \(2p^2q\) by \(4pq^4\): \[ 2p^2q imes 4pq^4 = 8p^3q^5 \]

Step 2: Combine All Terms

After distributing, write down all the results together to form a simplified expression:\[10p^3q^2 - 6p^5q^3 + 8p^3q^5\]This is the final simplified expression.

Key Concepts

Distributive Property

When dealing with polynomial multiplication, the distributive property is a crucial tool. This property lets you expand expressions in a systematic manner. In our example, the distributive property is used to multiply the term outside the parentheses, which is expressed as \(2p^2q\), across each term inside the parentheses separately.

Here's how the distributive property works:

• First, multiply \(2p^2q\) with the first term inside the parentheses: \(5pq\). The result is \(10p^3q^2\).
• Next, multiply \(2p^2q\) by the second term \(-3p^3q^2\). This gives \(-6p^5q^3\).
• Finally, multiply it by the third term \(4pq^4\), resulting in \(8p^3q^5\).

Each of these products is an individual expression, and when combined, they transform into a single expanded expression. Understanding the process of applying the distributive property is key to handling more complex algebraic expressions efficiently.

Simplification of Expressions

Simplifying expressions is the process of making a mathematical expression more compact and manageable. This involves combining like terms and arranging them in a simpler form. After distributing in the example given, you obtain a list of terms: \(10p^3q^2\), \(-6p^5q^3\), and \(8p^3q^5\).

Here's how you simplify these expressions:

• Identify any like terms. Like terms have the exact same variables raised to the same powers. In this example, you don't have like terms, but understanding this concept is crucial for other problems.
• Since there are no like terms here, the expression \(10p^3q^2 - 6p^5q^3 + 8p^3q^5\) is already in its simplest form.

An essential skill in algebra is recognizing when an expression is as simple as it gets, as in this example. Practice with simplifying expressions helps strengthen this skill.

Exponents in Algebra

Understanding exponents is vital when working with algebraic expressions. Exponents represent repeated multiplication of the same number or variable. In our polynomial multiplication task, exponents help compactly express multiplication.

Consider the examples in the solution:

• When multiplying \(2p^2q\) by \(5pq\), notice you add the exponents of \(p\) and \(q\) separately. So, \(p^2\) and \(p\) becomes \(p^{3}\), and \(q\) and \(q\) becomes \(q^2\).
• The same logic applies to other terms, where exponents of the same base (like \(p\) and \(q\)) add up accordingly, helping you to keep track of each variable in a simplified manner.

Remember that mastering exponents aids in handling larger expressions without getting overwhelmed. It also builds a foundation for more advanced mathematical concepts.