Question

Prove that \((A+B)(C+D)=A C+A D+B C+B D\)

Short answer

By using the distributive property, the expression expands to \(AC + AD + BC + BD\).

Step-by-step solution

Understand the expression

The expression \((A+B)(C+D)\)is a multiplication of two binomials. We need to expand it to show that it equals \(AC + AD + BC + BD\).

Apply the distributive property

To expand \((A+B)(C+D)\),we should use the distributive property: distribute each part of the first binomial\((A + B)\)to each part of the second binomial\((C + D)\).

Distribute A

Multiply\(A\)by each term in\(C + D\).This gives us:\(A \times C = AC\)and\(A \times D = AD\).

Distribute B

Similarly, multiply\(B\)by each term in\(C + D\).This results in:\(B \times C = BC\)and\(B \times D = BD\).

Combine the terms

Combine all the distributed terms obtained in the previous steps. The expanded expression will thus be:\(AC + AD + BC + BD\).

Final result

We have shown that the expression \((A+B)(C+D)\)expands to \(AC + AD + BC + BD\).This confirms the given statement.

Key concepts

Binomial Expansion

The binomial expansion is a method in algebra that simplifies expressions involving the multiplication of binomials. A binomial is an algebraic expression with two terms joined by a plus or minus sign, for example, \( A + B \) or \( C + D \). When we multiply two binomials together, like \( (A+B)(C+D) \), we aim to express it as a sum of simpler terms.

The primary goal is to apply the distributive property, which allows us to multiply each term in the first binomial by each term in the second one. This step-by-step method ensures that every combination of terms from each binomial is considered, resulting in four terms: \( AC, AD, BC, \) and \( BD \). This transformation of the product of binomials into a sum is known as expanding the binomial.

Mastering the binomial expansion is essential not just in algebra but also in calculus and other mathematical fields. By practicing these concepts, solving more complex problems becomes easier over time.

Algebraic Expressions

Algebraic expressions are the core language of algebra, comprising variables, numbers, and arithmetic operations such as addition, subtraction, multiplication, and division. Each expression can represent patterns, relationships, or quantities in a concise form.

An expression like \( A+B \) isn't just a random collection of letters and numbers; it handles specific roles:

• \( A \) and \( B \) represent variables or constants.
• The plus sign indicates an operation between them.

Understanding algebraic expressions is essential because they are the building blocks for equations and functions. They allow us to model real-world scenarios, solve problems, and make predictions based on relationships between variables.

The beauty of algebra is its ability to simplify and solve problems that initially seem complicated, all through these expressions. As you progress, manipulating these expressions becomes a handy tool for tackling a variety of mathematical problems.

Polynomial Multiplication

Polynomial multiplication extends the concept of multiplying binomials, but it can involve expressions with many terms, known as polynomials. When multiplying polynomials, the objective is to ensure that each term in one polynomial multiplies every term in the opposite polynomial, a process known as distributing.

For example, multiplying the binomials \( (A + B)(C + D) \) is a simplified case of polynomial multiplication. To solve this:

• Multiply \( A \) by \( C \) to get \( AC \).
• Multiply \( A \) by \( D \) to get \( AD \).
• Multiply \( B \) by \( C \) to obtain \( BC \).
• Multiply \( B \) by \( D \) to get \( BD \).

This expanded form of a polynomial multiplication makes each contribution explicit, resulting in a clearer, more manageable expression. When dealing with more than two binomials or higher degree polynomials, the same distributive process is applied, although it might involve more terms.

Gaining proficiency in polynomial multiplication is crucial for solving equations, simplifying expressions, and analyzing complex algebraic expressions efficiently. It also forms a foundation for understanding higher levels of mathematics, including calculus and linear algebra.