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Chapter 9: Problem 22

Expand and simplify each expression. $$ (4 k-7)(-2 k+3) $$

Short Answer

Expert verified
-8k^2 + 26k - 21

Step by step solution

01

Apply the Distributive Property

Use the distributive property to expand the expression (4k - 7)(-2k + 3). Multiply each term in the first binomial by each term in the second binomial.
02

Multiply the Terms

Multiply the terms individually: (4k)(-2k) = -8k^2, (4k)(3) = 12k, (-7)(-2k) = 14k, (-7)(3) = -21.
03

Combine Like Terms

After performing the multiplications, combine the like terms to simplify the expression: -8k^2 + 12k + 14k - 21.
04

Simplify the Expression

Combine the like terms 12k and 14k: -8k^2 + 26k - 21.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property
To solve algebraic expressions, one essential technique is using the distributive property. This property helps you expand expressions efficiently. The distributive property states that for any numbers or variables a, b, and c, the following holds: a(b + c) = ab + ac. In our example, we applied the distributive property to (4k - 7)(-2k + 3). This means we need to multiply every term in the first binomial (4k and -7) by every term in the second binomial (-2k and 3). It's the foundational step for solving more complex algebraic expressions.
combining like terms
Once you have expanded an expression using the distributive property, the next step is to simplify it by combining like terms. Like terms are terms that have the same variables raised to the same power. Only the coefficients (the numbers in front of the variables) are different. In our example, after multiplication, we get: -8k^2, 12k, 14k, and -21. The like terms here are 12k and 14k. We combine them to get: -8k^2 + (12k + 14k) - 21, which simplifies to -8k^2 + 26k - 21. Combining like terms helps to keep the expression simple and tidy.
multiplying binomials
Multiplying binomials involves using the distributive property, often described by the FOIL method. FOIL stands for First, Outer, Inner, Last, referring to multiplying terms in two binomials in a specific order: First: Multiply the first terms in each binomial. Outer: Multiply the outer terms in the binomials. Inner: Multiply the inner terms in the binomials. Last: Multiply the last terms in each binomial. In our example (4k - 7)(-2k + 3), First: (4k * -2k = -8k^2), Outer: (4k * 3 = 12k), Inner: (-7 * -2k = 14k), Last: (-7 * 3 = -21) Multiplying binomials helps break down the task of expansion into manageable steps.
algebraic expressions
Algebraic expressions contain numbers, variables, and operations, such as addition, subtraction, multiplication, and division. Understanding these expressions and knowing how to manipulate them is fundamental in algebra. For example, in the expression (4k - 7)(-2k + 3), 4k, -7, -2k, and 3 are terms. The expression shows how combining numbers with variables can form more complex expressions. By mastering the techniques of expanding and simplifying algebraic expressions, you open the door to solving more complicated algebra problems. This involves leveraging properties like the distributive property, combining like terms, and systematic methods such as FOIL.

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Most popular questions from this chapter

Simplify each expression. $$ \frac{7}{k-2}-\frac{5}{2(k-2)} $$

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