Problem 47 - Algebra 1: Concepts and Skills

Question

$$ (9 w-5)(7 w-12) $$

Step-by-Step Solution

Answer

The simplified expression is \(63w^2 - 143w + 60\)

Step 1: Distribute the first term of the first expression

Multiply the first term of the first expression \(9w\) with all terms in the second expression: \(9w * 7w = 63w^2\) and \(9w * -12 = -108w\)

Step 2: Distribute the second term of the first expression

Multiply the second term of the first expression \(-5\) with all terms in the second expression: \(-5 * 7w = -35w\) and \(-5 * -12 = 60\)

Step 3: Combine like terms

The resulting expression after distributing is \(63w^2 - 108w - 35w + 60\). We can simplify it by combining the like terms \(-108w\) and \(-35w\) to get \(63w^2 - 143w + 60\)

Key Concepts

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, connected by addition, subtraction, multiplication, and non-negative integer exponents. They are foundational in algebra and can vary in complexity. In our exercise, we work with the polynomial

• The term "\(9w\)" is a monomial (a single term polynomial) consisting of the variable \(w\) with a coefficient of 9.
• Similarly, "-5" is also a monomial.

When these terms are part of an expression within parentheses -- like \((9w - 5)\) and \((7w - 12)\) -- they become binomials, meaning these polynomials have exactly two terms. Understanding these terms helps break down mathematical problems and paves the way for operations like addition, subtraction, and multiplication of those polynomials.

Remember, a polynomial can have more than just one or two terms, but at their simplest, like here, they often appear first as either single or double-term structures. The operations performed typically introduce more terms and require further simplification.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions, often helping make complex problems more manageable. "Like terms" in algebra mean terms within an expression that have the same variables raised to the same power, making them comparable.

In our example, after distributing all terms, the expression was

• \(63w^2 - 108w - 35w + 60\).

The terms \(-108w\) and \(-35w\) are like terms because they both contain \(w\) raised to the power of one. To combine these, we add the coefficients:

• \(-108 - 35\), resulting in \(-143w\).

By simplifying, we express the given expression in a simpler form, leading to a positive outcome when dealing with calculations and further algebraic operations.

Simplification through combining like terms often reduces the complexity of an expression, which helps guide equations towards solutions more efficiently.

Algebraic Multiplication

Algebraic multiplication involves distributing terms across an expression following common rules of arithmetic but applying them to variables and coefficients. This process was applied in the given exercise using the distributive property to multiply every term inside one polynomial by each term in another polynomial.

Here's a breakdown using the example:

• First, distribute \(9w\) from \((9w - 5)\) to every term in the expression \((7w - 12)\). The results are: \(9w \times 7w = 63w^2\) and \(9w \times -12 = -108w\).
• Next, distribute the second term, \(-5\), from \((9w - 5)\) with each term in \((7w - 12)\), resulting in: \(-5 \times 7w = -35w\) and \(-5 \times -12 = 60\).

Each step involves applying multiplication one pair of terms at a time, ensuring each variable and coefficient are adequately accounted for before moving on.

This systematic approach to algebraic multiplication allows for clear, step-by-step simplification, leading to a concise and easy-to-understand expression in the solution.