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Chapter 4: Problem 2

The expression \(4 x^{3}-5 x^{2}\) has exactly (one / two / three) term(s).

Short Answer

Expert verified
Two terms.

Step by step solution

01

Understand the Expression

The expression given is \( 4x^3 - 5x^2 \). This expression is a polynomial.
02

Identify Each Term

A term in a polynomial is a part that is added or subtracted. In the expression \(4x^3 - 5x^2\), the plus or minus signs help in identifying separate terms.
03

Count the Terms

In the expression \(4x^3 - 5x^2\), there are two separate parts: \(4x^3\) and \(-5x^2\). Therefore, there are two terms in this expression.

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Key Concepts

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

Identifying Terms in a Polynomial
A polynomial is an algebraic expression that consists of variables, coefficients, and exponents. To identify terms in a polynomial, you need to look for parts of the expression separated by plus (+) or minus (-) signs. These signs indicate the boundaries between terms.

For example, in the expression \(4x^3 - 5x^2\), the terms are \(4x^3\) and \(-5x^2\). Each term includes a coefficient (like 4 or -5) and a variable part (like \(x^3\) or \(x^2\)).

Remember: Terms can include just a number (like 5) or a combination of numbers and variables (like \(3x^2\)). The key is to look for the plus or minus signs to find each distinct term.
Polynomial Structure
Understanding the structure of a polynomial is essential for identifying its components and solving problems effectively. A polynomial is composed of terms, each with a coefficient, variable, and possibly an exponent.

Here is how you can break down a polynomial:
  • Coefficients: These are the numerical factors that multiply the variables (e.g., 4 in \(4x^3\)).
  • Variables: These are symbols representing quantities that can change (e.g., x in \(4x^3\)).
  • Exponents: These indicate how many times the variable is multiplied by itself (e.g., 3 in \(x^3\)).

A polynomial can have one or more terms. For instance, in the expression \(4x^3 - 5x^2\), it has two terms: \(4x^3\) and \(-5x^2\). The overall structure can be more complex with multiple terms summed or subtracted, but the fundamental components remain the same.
Counting Terms in an Expression
Counting the number of terms in a polynomial is straightforward once you know how to identify them. Simply count each part of the expression that is separated by plus or minus signs.

For instance, in the polynomial \(4x^3 - 5x^2\), the terms are separated by a minus sign (-). Hence, you can identify two distinct terms: \(4x^3\) and \(-5x^2\).

  • Look for the + or - signs.
  • Each section between the signs is one term.

It鈥檚 important to note that a term can be positive or negative depending on the sign in front of it. In more complex polynomials, you might see more terms, but the steps to count them remain the same. Just keep an eye out for those operation signs!

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Most popular questions from this chapter

Multiply each of the following. (a) \((x-1)\left(x^{2}+x+1\right)\) (b) \((y-2)\left(y^{2}+2 y+4\right)\) Describe the final product as "the ___________of two_______."

$$\text { Subtract the sum of } 9 t^{3}-3 t+8 \text { and } t^{2}-8 t+4 \text { from the sum of } 12 t+8 \text { and } t^{2}-10 t+3$$

Without actually performing the operations, mentally determine the coefficient of the \(x\) -term in the simplified form of $$\left(-8 x^{2}-3 x+2\right)-\left(4 x^{2}-3 x+8\right)-\left(-2 x^{2}-x+7\right)$$

In \(2016, \$ 1.14 \times 10^{10}\) was spent to attend motion pictures in the United States and Canada. The total number of tickets sold was 1.32 billion. What was the average ticket price (to the nearest cent) for a movie? (Data from Motion Picture Association of America.)

Graph each equation by completing the table of values. $$ \begin{aligned} &y=2 x^{2}+2\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$
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