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Chapter 6: Problem 118

Perform the indicated operation. $$(x+2)\left(x^{2}-2 x+4\right)$$

Short Answer

Expert verified
The simplified form of the expression is \(x^{3}\).

Step by step solution

01

Distribute the first term

First, multiply the term \(x\) from the binomial, \(x+2\), to each term in the trinomial, \(x^{2}-2 x+4\). This gives \(x * x^{2}- x * 2x + x * 4\), which simplifies to \(x^{3} - 2x^{2} + 4x\).
02

Distribute the second term

Next, multiply the constant term from the binomial, which is 2, to each term in the trinomial. This gives \(2 * x^2 - 2 * 2x + 2 * 4\), which simplifies to \(2x^{2} - 4x + 8\).
03

Combine like terms

Finally, add the two resulting trinomials together, combining terms with the same exponent. This gives \((x^{3} - 2x^{2} + 4x) + (2x^{2} - 4x + 8)\), which simplifies to \(x^{3}\).

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Key Concepts

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

Polynomials
Polynomials are mathematical expressions consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. They are a fundamental component in algebra, forming the backbone for more complex calculations. A simple polynomial might look like this: \(5x^3 - 3x + 2\). Here, each part of the expression such as \(5x^3\), \(-3x\), and \(2\), is called a "term."

Key features of polynomials include:
  • The highest power of the variable, called the "degree," determines the behavior and classification of the polynomial. For example, in \(5x^3\), the degree is 3.
  • Polynomials can have multiple terms, or just one; for instance, \(x\) is a polynomial with one term.
  • The coefficients (numbers in front of variables, such as 5 and 3 in the example above) tell us the "weight" of each term.
  • They can be categorized as monomials (one term), binomials (two terms), trinomials (three terms), and so on.
Distributive Property
The distributive property is a useful algebraic principle that simplifies expressions and equations involving polynomials. It allows you to multiply a term across a sum inside parentheses. This is crucial for breaking down expressions like binomials and trinomials to perform algebraic operations.

To apply the distributive property, consider \((a+b) \times c\). You would do the following steps:
  • Multiply \(a\) by \(c\), resulting in \(ac\).
  • Multiply \(b\) by \(c\), resulting in \(bc\).
  • Add up both results to get \(ac + bc\).
This property allows simplifying expressions efficiently by eliminating parentheses and preparing the terms for further operations such as combining like terms.
Combining Like Terms
Combining like terms is an essential skill in simplifying polynomial expressions. "Like terms" are terms that contain the same variables raised to the same power. When you combine them, you essentially add or subtract coefficients that accompany the variable. For example, in the expression \(3x^2 + 5x - 2x^2 + 4x\), the terms \(3x^2\) and \(-2x^2\) are like terms, and so are \(5x\) and \(4x\).

Steps to combine like terms:
  • Identify the terms with the same variable and exponent. Group them together.
  • Add or subtract the coefficients of these like terms.
  • Rewrite the polynomial using the combined terms.
In our example, after combining like terms, it simplifies to \(1x^2 + 9x\). This process is crucial for simplifying and solving polynomial expressions efficiently.

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Most popular questions from this chapter

Factor: \(9 x^{2}-16 .\)

You are about to take a great picture of fog rolling into San Francisco from the middle of the Golden Gate Bridge, 400 feet above the water. Whoops! You accidently lean too far over the safety rail and drop your camera. The height, in feet, of the camera after \(t\) seconds is modeled by the polynomial \(400-16 t^{2} .\) The factored form of the polynomial is \(16(5+t)(5-t) .\) Describe something about your falling camera that is easier to see from the factored form, \(16(5+t)(5-t),\) than from the form \(400-16 t^{2}\)

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$-7 y^{3}+21 y^{2}-14 y$$

Factor completely. $$(x-6)^{2}-9 a^{2}$$

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. It takes a great deal of practice to get good at factoring a wide variety of polynomials.
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