Algebra: Polynomials and Their Degrees
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Questions and Answers

What is a monomial?

  • An expression with two terms.
  • An expression with multiple coefficients.
  • An expression with three terms. (correct)
  • An expression with one term. (correct)

Which of the following correctly defines a binomial?

  • An expression with three terms.
  • An expression with no terms.
  • An expression that has one term.
  • An expression that has two terms. (correct)

What characterizes a trinomial?

  • It is an expression with two variables.
  • It is an expression with only one variable.
  • It has three terms. (correct)
  • It contains only coefficients.

What does the term 'polynomial' refer to?

<p>An expression made up of variables and coefficients. (B)</p> Signup and view all the answers

In the expression 38m, what is 38 considered?

<p>A coefficient. (C)</p> Signup and view all the answers

What is an expression in mathematics?

<p>A mathematical phrase containing numbers, variables, and operators. (B)</p> Signup and view all the answers

Which of the following is NOT a term in the expression 7a² - $ rac{3}{5}ab + 6b²?

<p>4a + b (D)</p> Signup and view all the answers

How many terms are in the expression 3x + 9?

<p>Two (B)</p> Signup and view all the answers

Which of the following correctly identifies a term in the expression -7y + $ rac{1}{2}$?

<p>-7y (A)</p> Signup and view all the answers

Which of the following is an example of an expression?

<p>3x + 9 (C)</p> Signup and view all the answers

What is the degree of the polynomial -3x² + 4x + 7?

<p>2 (A)</p> Signup and view all the answers

Which term represents a constant in the expression 8x - 4 + 3y?

<p>-4 (B)</p> Signup and view all the answers

What is the standard form of the expression 5 + 2x³ - x?

<p>2x³ - x + 5 (C)</p> Signup and view all the answers

How do you identify the degree of the polynomial 10y⁴ - y + 6?

<p>By locating the term with the largest exponent (D)</p> Signup and view all the answers

What is the degree of the constant 0.5 in polynomial terms?

<p>0 (A)</p> Signup and view all the answers

What is the degree of the monomial $7a^2$?

<p>2 (C)</p> Signup and view all the answers

Calculate the degree of the monomial $1.79x^3y^2$.

<p>5 (B)</p> Signup and view all the answers

What is the degree of the monomial $\frac{3}{5}abc^2$?

<p>7 (C)</p> Signup and view all the answers

What is the implied exponent of a variable without a written exponent in a monomial?

<p>1 (C)</p> Signup and view all the answers

Which of the following expressions has the highest degree?

<p>$5y^3$ (C)</p> Signup and view all the answers

What is the first term written when rewriting the given polynomial expression in standard form?

<p>(2/9)a²bc⁸ (C)</p> Signup and view all the answers

Which variable comes first when arranging the polynomial terms in alphabetical order?

<p>a (B)</p> Signup and view all the answers

In the final standard form, what is the coefficient of the term 4a³b²c⁵?

<p>4 (C)</p> Signup and view all the answers

What is the last term written in the standard form of the polynomial?

<p>7abc² (D)</p> Signup and view all the answers

What is the highest power of 'a' in the expression before rewriting it in standard form?

<p>3 (C)</p> Signup and view all the answers

What is the first step in rewriting the polynomial 9x² - 4x + 5/2 + 10x³ in standard form?

<p>Identify the term with the highest exponent (A)</p> Signup and view all the answers

How should terms be arranged in a polynomial with two variables, such as 7x³y⁵ - 8x²y²?

<p>First alphabetically by variable, then in descending order of exponents for the leading variable (C)</p> Signup and view all the answers

In the standard form of a polynomial, what should be done with the signs of the terms?

<p>Maintain the original signs of the terms (C)</p> Signup and view all the answers

For the polynomial expression 10x³ + 9x² - 4x + 5/2, what is the correct standard form arrangement?

<p>10x³ + 9x² - 4x + 5/2 (B)</p> Signup and view all the answers

In which order should the terms in a polynomial be organized when rewriting it?

<p>First by alphabetically and then in descending order of the leading variable's exponents (B)</p> Signup and view all the answers

Flashcards

Monomial definition

An expression with only one term.

Binomial example:

An expression with two terms

Trinomial

Expression with three terms.

Polynomial definition

Expression with many terms.

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Coefficient

Numerical part of a term.

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What is an expression?

A mathematical phrase that combines numbers, variables, and operators (like +, -, ×, and ÷).

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What is a term?

A single part of an expression. It can be a number, a variable, or a number multiplied by a variable(s).

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Example of a term

A number by itself, or a variable multiplied by a number. Examples: 3x, 9, -7y, 38m.

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How are terms separated?

Terms are separated by plus (+) or minus (-) signs in an expression.

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"3x + 9" - How many terms?

This expression has two terms: 3x and 9.

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Degree of a Polynomial

The largest exponent of the variable in a polynomial.

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Constant in an Expression

A number that doesn't change.

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Degree of a Constant

0

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Standard Form of an Expression

Writing terms in decreasing order of exponents, ending with the constant.

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Example of Standard Form

Writing 3 + 7y² in Standard Form as 7y² + 3

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Monomial Degree

The sum of the exponents of the variables in a monomial.

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Monomial Exponent

The power to which a variable is raised.

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Implied Exponent

A variable without a written exponent has an exponent of 1.

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Degree Example

Finding the degree of a monomial by summing exponents of variables. Like 3a^2b^5 has a degree of 2 + 5 = 7.

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Standard Form of a Polynomial

A polynomial written with terms arranged in descending order of their exponents, starting with the term with the highest exponent.

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Rewriting Polynomials with Multiple Variables

First, arrange terms alphabetically based on the variables. Then, arrange the terms in descending order of exponents for the leading variable alphabetically first.

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Polynomial Example: 7x³y⁵ - 8x²y²

The highest exponent of x is 3 in 7x³y⁵, so it comes first. Then, -8x²y² with an x exponent of 2.

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Maintaining Signs in Polynomials

When rearranging a polynomial, keep the original signs (+ or -) in front of each term.

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Rewriting Polynomials in Standard Form

The process of identifying the term with the highest exponent and arranging terms in descending order of exponents.

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Leading Variable

The variable appearing first in the alphabetical order of the terms in a polynomial expression.

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Why Rewrite in Standard Form?

It helps simplify polynomial expressions by organizing them according to the power of their variables. It makes it easier to compare and combine terms.

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How to Find the Leading Term

Identify the term with the largest exponent of the variable. In a polynomial with multiple variables, the highest exponent of the leading variable determines the leading term.

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Coefficients in Standard Form

The numerical part of each term in the standardized polynomial. They are usually written in front of the variables and their exponents.

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