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Chapter 1: Problem 156

Perform the indicated operation or operations. $$(2 x-1)\left(x^{2}+x-2\right)$$

Short Answer

Expert verified
The final answer to the indicated operation is \(2x^3 + x^2 - 5x + 2\).

Step by step solution

01

Distribute \(2x\) Across Trinomial

Multiply \(2x\) by each term in \(x^2+x-2\). Firstly, \(2x\) * \(x^2\) gives \(2x^3\). Secondly, \(2x\) * \(x\) gives \(2x^2\). Finally, \(2x\) * \(-2\) gives \(-4x\)
02

Distribute \(-1\) Across Trinomial

Multiply \(-1\) by each term in \(x^2+x-2\). Firstly, \(-1\) * \(x^2\) gives \(-x^2\). Secondly, \(-1\) * \(x\) gives \(-x\). Finally, \(-1\) * \(-2\) gives \(2\)
03

Add Results

Sum together all results found in steps 1 and 2 which yields the final polynomial: \(2x^3 + 2x^2 - 4x - x^2 - x + 2\). There are like terms in this polynomial, which are \(2x^2\) and \(-x^2\), and \(-4x\) and \(-x\). Combine these like terms by adding the coefficients (numbers in front of the variables).

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Key Concepts

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

Distributive Property
When multiplying polynomials, it's essential to understand the distributive property. This property allows you to multiply a term outside the parentheses by each term inside the parentheses, effectively 'distributing' the multiplier. Imagine you're handing out snacks at a party; the distributive property ensures that everyone inside the parentheses gets their share of snacks (in our case, the multiplication).

Take, for example, the expression \(2x(x^2 + x - 2)\). According to the property, you would perform the following steps:
  • Multiply \(2x\) with \(x^2\) to get \(2x^3\).
  • Then, multiply \(2x\) with \(x\) to obtain \(2x^2\).
  • Finally, \(2x\) multiplied by \(\-2\) equals \(\-4x\).
Each term within the parentheses has been multiplied by \(2x\), illustrating the distributive property in action.
Combining Like Terms
After distributing, you often end up with an expanded polynomial that may contain 'like terms.' These are terms that are identical in their variable parts, differing only in their coefficients. Combining like terms is like tidying up a room by grouping similar items together.

For example, in the polynomial \(2x^3 + 2x^2 - 4x - x^2 - x + 2\), there are two pairs of like terms: \(2x^2\) and \(\-x^2\), and \(\-4x\) and \(\-x\). To combine them:
  • Add \(2x^2\) and \(\-x^2\) to get \(x^2\).
  • Then, add \(\-4x\) and \(\-x\) to obtain \(\-5x\).
Combining them simplifies the polynomial to \(2x^3 + x^2 - 5x + 2\), making it easier to understand and use.
Multiplying Binomials
Multiplying binomials involves using both the distributive property and combining like terms efficiently. A binomial is a polynomial with two terms, such as \(2x - 1\). When multiplying two binomials, say \(2x - 1\) and \(x^2 + x - 2\), you distribute each term in the first binomial across the second.

Here's a step-by-step guide:

  • Multiply \(2x\) by each term in \(x^2 + x - 2\).
  • Do the same with the \(\-1\), multiplying it by each term in \(x^2 + x - 2\).
After distributing, you'll combine like terms to tidy up the polynomial and complete the multiplication process. This systematic approach to multiplication can also be extended to trinomials and higher-order polynomials.

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Most popular questions from this chapter

In Exercises \(53-64,\) complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+12 x-6 y-4=0$$

In Exercises \(67-70,\) graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned} x^{2}+y^{2} &=9 \\ x-y &=3 \end{aligned}$$

In Exercises \(53-64,\) complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+x+y-\frac{1}{2}=0$$

What must be done to a function's equation so that its graph is shifted vertically upward?

Exercises \(101-103\) will help you prepare for the material covered in the next section. Write an algebraic expression for the fare increase if a \(\$ 200\) plane ticket is increased to \(x\) dollars.
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