Problem 9 - Algebra 1: Concepts and Skills

Question

Use the distributive property to find the product. $$ (4 x+7)(-2 x) $$

Step-by-Step Solution

Answer

The product is \(-8x^2 - 14x\).

Step 1: Distribute the Monomial to Each Term in the Binomial

To start, distribute \(-2x\) to \(4x\) and \(7\). This is done by multiplying \(-2x\) with each term separately. It will look like this: \((-2x * 4x) + (-2x * 7)\).

Step 2: Multiply the Terms

Now, perform the multiplication for each term. Remember, when you multiply two terms which both have an \(x\), you add their powers (in this case, their power is 1). So, \(-2x * 4x = -8x^2\). And \(-2x * 7 = -14x\).

Step 3: Simplify the Expression

Combining the two terms from the previous step gives the final simplified expression: \(-8x^2 - 14x\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \( x \)), and operations such as addition, subtraction, multiplication, and division. They do not include an equal sign, unlike equations. In algebraic expressions, different elements come together to form terms. A term is a product of numbers and variables. For example, in the expression \( 4x + 7 \), \( 4x \) and \( 7 \) are both terms.

Here are some key points to remember about algebraic expressions:

• **Coefficients**: These are the numerical parts of the terms, like the \( 4 \) in \( 4x \).
• **Variables**: Letters that represent unknown values, such as \( x \) in our expression.
• **Constants**: Numbers without a variable component, like \( 7 \) in \( 4x + 7 \).

Understanding these components helps in simplifying expressions and solving algebraic problems effectively.

Multiplying Polynomials

Multiplying polynomials is an important skill in algebra that involves applying the distributive property. The distributive property allows us to multiply each term in one polynomial by each term in another polynomial, ensuring that everything combines correctly.

In our exercise, we are multiplying a binomial, \( (4x + 7) \), by a monomial, \( -2x \). The distributive property guides us in breaking down the multiplication process as follows:

• Distribute \(-2x\) to \(4x\), resulting in \( -2x \times 4x \).
• Distribute \(-2x\) to \(7\), resulting in \( -2x \times 7 \).

By using the properties of exponents during multiplication, \( -2x \times 4x \) becomes \(-8x^2\), since the powers of \( x \) add up. Similarly, \(-2x \times 7\) simplifies to \(-14x\). The result is an expression where the variable's degree (the highest power of \( x \)) helps in determining the polynomial's classification.

Simplification in Algebra

Simplification in algebra involves reducing an expression to its simplest form. This process involves combining like terms, factoring, and using basic arithmetic operations to make the expression as straightforward as possible.

In the step-by-step solution provided, after using the distributive property to expand \( (4x + 7)(-2x) \), we achieved two terms: \(-8x^2\) and \(-14x\). These terms form a simplified expression because:

• The terms share no similar power of \( x \) (\(-8x^2\) is different from \(-14x\)).
• Each term is expressed in its lowest possible terms, given the multiplication performed.

Through simplification, expressions become easier to interpret and solve in equations, enabling clearer communication of mathematical ideas.