Problem 46 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ -7(4 x+2) $$

Step-by-Step Solution

Answer

Question: Simplify the expression: -7(4x + 2) Answer: -28x - 14

Step 1: Apply Distributive Property of Multiplication

Distribute the scalar -7 across the terms inside the parenthesis. $$ -7(4x+2) = -7(4x) + (-7)(2) $$

Step 2: Perform Multiplication on Each Term

Multiply each term by -7: $$ -7(4x) + (-7)(2) = -28x - 14 $$

Step 3: Combine Like Terms (If Necessary)

In this expression, there are no like terms to combine, so the final answer is: $$ -28x - 14 $$

Key Concepts

Combining Like Terms

Combining like terms is a technique used in algebra to simplify expressions or equations. Like terms are terms that have exactly the same variable factors. These are terms where the variables and their exponents are identical. For instance, in the expression \(3x + 5x\), the terms \(3x\) and \(5x\) are like terms because both have the variable \(x\) to the first power. To combine them, you simply add or subtract their coefficients, resulting in \(8x\).

However, in the given exercise \(-7(4x + 2) = -28x - 14\), there are no like terms after multiplication, thus no combining is necessary. When working with algebraic expressions, always look for and combine like terms to simplify the expression as much as possible. This process is essential for solving equations and understanding polynomial expressions.

Multiplying Polynomials

Multiplying polynomials involves using the distributive property to multiply each term of one polynomial by each term of another. This method, also known as the FOIL method when dealing with binomials, stands for First, Outer, Inner, Last, and it refers to the position of the terms in each binomial.

In the case of the exercise provided, we multiply a monomial \(-7\) by a binomial \(4x + 2\). Here, the distributive property is used to multiply the scalar \(-7\) by each term in the polynomial \(4x + 2\). The process of multiplication will give new terms that can be alike or unlike. In this instance, multiplication results in \(-28x - 14\), which are unlike terms and therefore do not require further simplification by combining. In more complex examples with more terms or higher degree polynomials, you would follow the same principle but do more multiplication and combining of like terms to reach a simplified form.

Applying Scalar Multiplication

Scalar multiplication in algebra involves multiplying a number (called a scalar) by every term within a polynomial. The scalar multiplies each coefficient in the polynomial, but does not change the variable or its exponents.

For our exercise, the scalar \(-7\) multiplies each term in the polynomial \(4x + 2\), which is done as so: \(-7 \times 4x\) and \(-7 \times 2\). It's important to note that scalar multiplication is distributive over addition in polynomials. This means that multiplying a scalar by a sum of terms is the same as multiplying every term by that scalar and then summing the results. This is applied in the step-by-step solution in which the scalar multiplication yielded \(-28x - 14\). Applying scalar multiplication correctly is vital for operations such as scaling figures in geometry, solving systems of equations, and understanding the basis in linear algebra.