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Chapter 0: Problem 54

Multiply the polynomials using the FOIL method. Express your answer as a single polynomial in standard form. $$ (x+4)(x-2) $$

Short Answer

Expert verified
The result is \(x^2 + 2x - 8\).

Step by step solution

01

Apply the FOIL method

The FOIL method stands for multiplying the First, Outer, Inner, and Last terms of each binomial. Identify the binomials as (x + 4) and (x - 2).
02

Multiply the First terms

First, multiply the first terms of the binomials: \(x \times x = x^2\).
03

Multiply the Outer terms

Next, multiply the outer terms: \(x \times (-2) = -2x\).
04

Multiply the Inner terms

Then, multiply the inner terms: \(4 \times x = 4x\).
05

Multiply the Last terms

Finally, multiply the last terms: \(4 \times (-2) = -8\).
06

Combine like terms

Combine all the terms we obtained from the previous steps: \(x^2 - 2x + 4x - 8\).
07

Simplify the expression

Combine the like terms \(-2x\) and \(4x\): \(-2x + 4x = 2x\). Thus, the polynomial simplifies to: \(x^2 + 2x - 8\).

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Key Concepts

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

Combining Like Terms
continued
like terms here are: \(-2x \, and\ 4x \) \(-2x + 4x = 2x \)
Combining like terms helps us to simplify our polynomial expression effectively.
Standard Form of a Polynomial
The standard form of a polynomial is a way of writing the polynomial with terms in descending order of their degree, which is the highest power of the variable.
  • This means that the term with the highest exponent should come first, followed by the next highest, and so on.
  • In our problem, the final simplified form is \(x^2 + 2x - 8\). As you can see, \( x^2 \) (degree 2) is first, followed by \( 2x \) (degree 1), and then \(-8 \) (degree 0).

Writing polynomials in standard form helps maintain a consistent structure and makes them easier to read and understand.

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Most popular questions from this chapter

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{x+h}+\sqrt{x-h}}{\sqrt{x+h}-\sqrt{x-h}}$$

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt{8 x^{3}}-3 \sqrt{50 x}$$

Simplify each expression. $$16^{-3 / 2}$$

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{2 x\left(1-x^{2}\right)^{1 / 3}+\frac{2}{3} x^{3}\left(1-x^{2}\right)^{-2 / 3}}{\left(1-x^{2}\right)^{2 / 3}} \quad \neq-1, x \neq 1$$

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}}$$
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