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Chapter 5: Problem 58

Find each product. Use the FOIL method. $$ (6-5 x)(2+x) $$

Short Answer

Expert verified
The product is \[12 - 4x - 5x^2.\]

Step by step solution

01

Understand the FOIL Method

The FOIL method helps in multiplying two binomials. FOIL stands for First, Outer, Inner, Last.
02

Multiply the First Terms

Multiply the first terms of each binomial: \[6 \times 2 = 12 \]
03

Multiply the Outer Terms

Multiply the outer terms: \[6 \times x = 6x \]
04

Multiply the Inner Terms

Multiply the inner terms: \[(-5x) \times 2 = -10x \]
05

Multiply the Last Terms

Multiply the last terms: \[(-5x) \times x = -5x^2 \]
06

Combine Like Terms

Combine all of the terms: \[12 + 6x - 10x - 5x^2 \]Simplify by combining like terms: \[12 - 4x - 5x^2 \]

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Key Concepts

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

Binomials
When we talk about binomials in algebra, we're referring to polynomials that have only two terms. Each term can be a constant, a variable, or the product of a constant and a variable.

For example, in the expression \(6 - 5x\), we have two terms: \(6\) and \(-5x\).
Similarly, in the expression \(2 + x\), we have two terms: \(2\) and \(x\).

Understanding binomials is the foundation for using the FOIL method, which we'll cover next.
  • Binomials have two parts.
  • They can appear in different forms but follow the general structure of \(a + b\) or \(a - b\).
Multiplication
Multiplication is a fundamental operation in algebra. When multiplying binomials, the FOIL method is very useful.

FOIL stands for First, Outer, Inner, and Last. This acronym helps us remember the order in which we multiply terms:
  • First: Multiply the first terms of each binomial.
  • Outer: Multiply the outer terms of each binomial.
  • Inner: Multiply the inner terms of each binomial.
  • Last: Multiply the last terms of each binomial.
Let's use the given exercise to demonstrate this.

Start with \( (6-5x)(2+x) \). Applying FOIL, we get:
  • First terms: \( 6 \times 2 = 12 \)
  • Outer terms: \( 6 \times x = 6x \)
  • Inner terms: \( -5x \times 2 = -10x \)
  • Last terms: \( -5x \times x = -5x^2 \)
This breakdown simplifies multiplication and helps keep calculations organized.
Combining Like Terms
After multiplying the binomials, you'll end up with several terms that need simplifying. This is where combining like terms comes into play. Like terms are those that contain the same variable raised to the same power.

For example, in the expression \( 12 + 6x - 10x - 5x^2 \), we have:
  • Constant term: \( 12 \)
  • \( x \) terms: \( 6x \) and \( -10x \)
  • \

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Most popular questions from this chapter

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 103 \times 97 $$

Perform each division. $$ \frac{3 t^{4}+5 t^{3}-8 t^{2}-13 t+2}{t^{2}-5} $$

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(2 y^{-1} z^{2}\right)^{2}\left(3 y^{-2} z^{-3}\right)^{3}}{\left(y^{3} z^{2}\right)^{-1}} $$

Multiply. $$ -3\left(x^{2}+7\right) $$

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(4 a^{2} b^{3}\right)^{-2}\left(2 a b^{-1}\right)^{3}}{\left(a^{3} b\right)^{-4}} $$
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