Chapter 4: Problem 30
s 27–32, fi nd the product of the binomials \((2 x+5)(x-2)(3 x+4)\)
Short Answer
Expert verified
The product of the binomials \((2x+5)(x-2)(3x+4)\) is \(6x^3+11x^2-26x-40\).
Step by step solution
01
First Distribution
Multiply each term in the first binomial, \((2x+5)\), with each term in the second binomial, \((x-2)\). This yields \(2x(x-2)+5(x-2)\) which simplifies to \(2x^2-4x+5x-10\). Combining like terms we get \(2x^2+x-10\). So, the product of the first two binomials is \(2x^2+x-10\).
02
Second Distribution
Multiply each term from the product of the first two binomials, \(2x^2+x-10\), with each term in the third binomial, \((3x+4)\). This yields \(2x^2(3x+4)+x(3x+4)-10(3x+4)\) which simplifies to \(6x^3+8x^2+3x^2+4x-30x-40\). Combining like terms we get \(6x^3+11x^2-26x-40\). So, the product of the three binomials is \(6x^3+11x^2-26x-40\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomials
A binomial is an algebraic expression that consists of exactly two terms. These terms are separated by either a plus or minus sign, making the expression simple yet powerful. Common examples include expressions like \(2x + 5\) and \(x - 2\). Binomials are fundamental elements in algebra because they often appear in various equations and problems.When dealing with binomials, you will frequently encounter operations like addition, multiplication, and factoring. In our exercise, we multiplied three binomials, which involves distributing each term across other terms in subsequent steps. Understanding binomials is crucial because it forms the basis for tackling polynomial problems.
Distributive Property
The Distributive Property is a foundational principle in algebra used to multiply a single term by two or more terms inside a parenthesis. It says that \(a(b + c) = ab + ac\). We employ the distributive property when we multiply binomials, as it allows us to distribute each term of one binomial over every term of another.For example, in the step-by-step solution for multiplying the binomials \((2x+5)\) and \((x-2)\), we distribute each term of the first binomial across every term of the second binomial:
- First, multiply \(2x\) by each term in \((x-2)\), giving \(2x^2 - 4x\).
- Next, multiply \(5\) by each term in \((x-2)\), giving \(5x - 10\).
Combining Like Terms
Combining like terms is a process where we add or subtract terms that have the same variable raised to the same power. It's a crucial step in simplifying algebraic expressions because it reduces the number of terms, making an expression easier to work with.For instance, in the simplification of \(2x^2 - 4x + 5x - 10\), we identify like terms by their variable component:
- \(-4x\) and \(+5x\) are like terms because they both have the variable \(x\) to the first power.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operation symbols like addition, subtraction, multiplication, and division. They form the building blocks of algebraic equations and problems.In the context of our exercise, we've been working with expressions like \(6x^3+11x^2-26x-40\). Such expressions can contain multiple terms with coefficients and variables raised to various powers. Understanding how to manipulate and combine these terms is key in solving algebra problems. When multiplying algebraic expressions, you’ll often use concepts like the distributive property and combining like terms to work through the problem efficiently. Mastering these techniques enables you to handle more complex polynomial expressions with confidence.