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Chapter 5: Problem 82

Multiply. $$ 4(x+7)(x-6) $$

Short Answer

Expert verified
The expanded expression is \(4x^2 + 4x - 168\).

Step by step solution

01

Identify the expression for multiplication

To multiply the expression, we must first identify the terms and their order. The expression given is \(4(x+7)(x-6)\). This consists of a constant factor \(4\) and two binomials \((x+7)\) and \((x-6)\). Start with the multiplication of the binomials.
02

Apply the distributive property to expand binomials

Expand \((x+7)(x-6)\) using the distributive property, also known as the FOIL (First, Outside, Inside, Last) method:- First: \(x \times x = x^2\)- Outside: \(x \times (-6) = -6x\)- Inside: \(7 \times x = 7x\)- Last: \(7 \times (-6) = -42\)Combine these to get: \[x^2 - 6x + 7x - 42\] Simplify the expression by combining like terms: \[x^2 + x - 42\]
03

Multiply the expanded polynomial by the constant

Next, multiply the expanded polynomial \((x^2 + x - 42)\) by the constant \(4\) using the distributive property. Perform the multiplication for each term:- \(4 \times x^2 = 4x^2\)- \(4 \times x = 4x\)- \(4 \times (-42) = -168\)Combine these results to form the final expression:\[4x^2 + 4x - 168\]

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Key Concepts

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

Binomials
Binomials are simply a type of polynomial with exactly two terms. They are usually written in the form \((a + b)\) or \((a - b)\), where "a" and "b" are any numbers or variables. In our exercise, the binomials are \((x + 7)\) and \((x - 6)\). These are crucial for understanding how different algebraic expressions interact when you multiply them together.
  • Each binomial consists of two distinct terms.
  • They are joined by either addition or subtraction.
  • In multiplying binomials, like in our example, we aim to find the product of all possible combinations of terms from each binomial.
Recognizing binomials is the first step in applying multiplication rules effectively, such as the Distributive Property and the FOIL Method.
Distributive Property
The Distributive Property is a fundamental algebraic property used to spread multiplication over addition or subtraction within an expression. In simpler terms, it allows us to take one factor outside of a parenthesis and multiply each term inside the parenthesis, like \(a(b + c) = ab + ac\). This property is used in expanding expressions.
  • Helps when multiplying multiple terms within a binomial.
  • Ensures every term gets multiplied correctly.
  • Is used in combination with the FOIL method when dealing with binomials.
In our solution, using the Distributive Property assured that every part of each binomial was considered in the expansion, and subsequently in the final multiplication by the constant.
FOIL Method
The FOIL Method is a specialized application of the Distributive Property, particularly designed for multiplying two binomials. It provides a framework for organizing the multiplication of each pair of terms:
  • First: Multiply the first term of each binomial.
  • Outside: Multiply the outer terms of the product expression.
  • Inside: Multiply the inner terms of the product expression.
  • Last: Multiply the last term of each binomial.
This method not only ensures accuracy but also makes it easier to visualize the multiplication process like in our example. After applying these steps to our binomials \((x+7)\) and \((x-6)\), we expanded and combined terms effectively. Subsequently, this was simplified by combining like terms, resulting in the polynomial \(x^2 + x - 42\) before multiplying by the constant 4.

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Most popular questions from this chapter

Is the product of a monomial and a monomial always a monomial? Explain.

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CHALLENGE PROBLEMS Simplify each expression. The variables represent natural numbers. A. \(x^{2 m} x^{3 m}\) B. \(\left(y^{5}\right)^{4}\) C. \(\frac{m^{8 x}}{m^{4 x}}\) D. \(\left(2 a^{6 y}\right)^{4}\)

Perform each division. $$ \frac{24 n^{12}}{8 n^{4}} $$
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