Problem 32 - Elementary Algebra

Question

\((9 w+5)(w-7)\)

Step-by-Step Solution

Answer

9w^2 - 58w - 35

Step 1 - Apply the Distributive Property

Use the distributive property of multiplication over addition to expand the expression. This means multiplying each term in the first parenthesis by each term in the second parenthesis.

Step 2 - Multiply Each Term

First, multiply the first terms: (9w) * (w) = 9w^2. Then, multiply the outer terms: (9w) * (-7) = -63w. Next, multiply the inner terms: (5) * (w) = 5w. Finally, multiply the last terms: (5) * (-7) = -35.

Step 3 - Combine Like Terms

Combine the terms that have the same variable part: 9w^2 + (-63w) + 5w + (-35) = 9w^2 - 58w - 35.

Key Concepts

Expanding Expressions

Expanding expressions involves using mathematical operations to break down more complex expressions into simpler components. In our exercise, we start with \( (9w + 5)(w - 7) \). To expand this, we use the distributive property. The distributive property states that for any three terms \(a\), \(b\), and \(c\): \[ a(b + c) = ab + ac \]. In the context of our exercise, this means multiplying each term in the first parenthesis by each term in the second parenthesis. First, distribute \(9w\) to both \(w\) and \(-7\). Then distribute \(5\) to both \(w\) and \(-7\). Here's the expanded form step-by-step: \[ (9w + 5)(w - 7) = (9w \cdot w) + (9w \cdot -7) + (5 \cdot w) + (5 \cdot -7) \]

Combining Like Terms

Combining like terms is a process where we simplify an expression by adding or subtracting the coefficients of terms that have the same variable parts. After expanding the expression \( (9w + 5)(w - 7) \), we get: \[ 9w^2 - 63w + 5w - 35 \] Now, look for terms that have the same variable. In this case, \( - 63w \) and \(+5w\) are like terms because they both have the variable \( w \). Combine these by adding their coefficients: \(-63 + 5 = -58\). Therefore: \[ -63w + 5w = -58w \] Incorporating this back into the expression, we get: \[ 9w^2 - 58w - 35 \]

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by each term in another polynomial, then combining like terms to simplify. For our given problem, we start with two binomials: \( (9w + 5) \) and \( (w - 7) \). The multiplication process consists of four steps: \begin{itemize} \item First Terms: \’ 9w \cdot w = 9w^2\ \item Outer Terms: \’ 9w \cdot -7 = -63w\ \item Inner Terms: \’ 5 \cdot w = 5w\ \item Last Terms: \’ 5 \cdot -7 = -35\ \end{itemize} Followed by combining the obtained results to achieve: \[ 9w^2 - 63w + 5w - 35 \] Those make polynomial multiplication an essential technique in algebra used to break down complex expressions into simple and manageable parts. The final simplified result, after combining like terms, is: \[ 9w^2 - 58w - 35 \]