Problem 18 - Beginning and Intermediate Algebra

Question

Multiply. $$5 p^{5} q^{2}\left(-5 p^{2} q+12 p q^{2}-p q+2 q-1\right)$$

Step-by-Step Solution

Answer

The short answer to the question is: \(-25p^7 q^3 + 60p^6 q^4 - 5p^6 q^3 + 10p^5 q^3 -\ 5p^5 q^2\)

Step 1: Apply Distributive Property

Multiply \(5p^5 q^2\) with each term in the parenthesis: \(5p^5 q^2(-5p^2q) + 5p^5 q^2(12pq^2) + 5p^5 q^2(-pq) + 5p^5 q^2(2q) + 5p^5 q^2(-1)\)

Step 2: Perform Multiplications and Simplify

Now, we need to perform the multiplication for each term and, when possible, simplify the exponents by adding them together: \((-25p^7 q^3) + (60p^6 q^4) + (-5p^6 q^3) + (10p^5 q^3) + (-5p^5 q^2)\)

Step 3: Combine Like Terms (if necessary)

In this case, there are no like terms to combine, so the solution remains: \(-25p^7 q^3 + 60p^6 q^4 - 5p^6 q^3 + 10p^5 q^3 - 5p^5 q^2\)

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions and perform multiplication across parentheses. It is represented by the formula:

• \(a(b + c) = ab + ac\)

This means that you multiply the term outside the parenthesis by each term inside the parenthesis.
In our exercise, we applied the distributive property by multiplying the term \(5p^5 q^2\) with each term inside the parenthesis:

• \( -5p^2q \)
• \( 12pq^2 \)
• \( -pq \)
• \( 2q \)
• \( -1 \)

This breaks down a complex multiplication task into manageable parts, simplifying our calculations.

The Process of Multiplying Polynomials

Multiplying polynomials involves multiplying each term in one polynomial by every term in another polynomial or expression. This process requires attention to both coefficients (numbers in front of the variables) and the variables themselves.
Here's how it looks for each term:

• Multiply the coefficients together. For instance, in \((-5 \times 5 = -25)\).
• Multiply the variable parts by adding their exponents, such as \(p^5 \times p^2 = p^{5+2} = p^7\).
• Do the same for the \(q\) terms, \(q^2 \times q = q^{2+1} = q^3\).

Performing these steps for every combination results in our fully expanded polynomial.

Adding Exponents of Like Bases

One of the key rules when working with polynomial multiplication is the exponent rule, where we add exponents when multiplying like bases.
Remember, the bases must be the same, like \(p\) and \(q\), to apply this rule.

• If multiplying \(p^a \times p^b\), then the result is \(p^{a+b}\).
• Similarly, \(q^a \times q^b = q^{a+b}\).

In our exercise, this was used to simplify terms such as \(p^5 \times p^2 = p^7\). This shows that when multiplying, you keep the base and add the exponents, ensuring accurate simplification.

Identifying and Combining Like Terms

Once polynomial multiplication is completed, identifying like terms is the next step. Like terms are those that have the same variable part, which in this context means identical base variables and exponents.
For example, terms like \(p^6q^3\) and \(p^6q^3\) would be like terms. They can be added or subtracted.However, in our exercise, no like terms existed in the resultant polynomial:

• \(-25p^7q^3 + 60p^6q^4 - 5p^6q^3 + 10p^5q^3 - 5p^5q^2\)

Ensuring that each term has a unique combination of exponent variables and coefficients verified that no further simplification was possible. This step is crucial to arriving at the most simplified form of the expression.