FNDMATH Lesson 1: Algebraic Expressions

Questions and Answers

What does a variable represent in mathematics?

An expression containing no numbers.

A fixed value in an expression.

A single letter that can take multiple values. (correct)

A constant that does not change.

Which of the following describes a binomial?

An expression with three terms.

An expression composed only of two terms. (correct)

A single term with no operation.

An expression that cannot be simplified.

What does the degree of a term refer to?

The highest exponent of any literal factor in that term. (correct)

The total number of variables in the expression.

The number of terms in an expression.

The value of the constant in an expression.

In the expression $3x^2y^3z$, which literal factor has the highest degree?

$y$ Signup and view all the answers

What indicates how many times a base occurs as a factor in a power?

The exponent. Signup and view all the answers

Which of the following is NOT a characteristic of a monomial?

It incorporates addition and subtraction. Signup and view all the answers

What is the result of $5x^3 imes 2y$?

$10x^3y$ Signup and view all the answers

What would be the result of dividing $-6x^{5}y^{-2}$ by $2x^{3}$?

$-3x^2y^{-2}$ Signup and view all the answers

In the context of algebraic expressions, which of the following represents a constant?

$3$ Signup and view all the answers

If $b^n$ is read as 'b raised to the nth power', what does $n$ represent?

The number of factors of the base. Signup and view all the answers

What is the result of $a^{m} \cdot a^{n}$ when $m$ and $n$ are positive integers?

$a^{m+n}$ Signup and view all the answers

Which expression simplifies to $12x^{2}z^{4}w^{-1}$ using the rule $\frac{a^{m}}{a^{n}} = a^{m-n}$?

$\frac{12x^{2}z^{4}w}{w^{2}}$ Signup and view all the answers

Applying the rule $(ab)^{m} = a^{m}b^{m}$, what is the correct expression for $(3xy)^{3}$?

$3^{3}x^{3}y^{3}$ Signup and view all the answers

What is the value of $\frac{(-27x^{5}z^{6}w^{7})}{12x^{2}z^{4}w}$ using the rule $\frac{a^{m}}{a^{n}} = a^{m-n}$?

$-\frac{9}{4}x^{3}z^{2}w^{6}$ Signup and view all the answers

Which of the following correctly applies the rule $\left(\frac{a}{b}\right)^{m} = \frac{a^{m}}{b^{m}}$ for $\left(\frac{x}{yz}\right)^{7}$?

$\frac{x^{7}}{y^{7}z^{7}}$ Signup and view all the answers

What is the simplified form of $(-2x^{2}y^{3}z)^{3}$ using $(a)^{n} = a^{n}$?

$-8x^{6}y^{9}z^{3}$ Signup and view all the answers

Using the rule $a^{m} = a^{m+n} \cdot a^{n}$, what is the result of $4w^{2}z^{3}(2wz-5w^{3}z^{2})$?

$8w^{3}z^{4} - 20w^{5}z^{5}$ Signup and view all the answers

What is the correct simplification of $a^{m} \cdot a^{n}$ if $m = 5$ and $n = 3$?

$a^{8}$ Signup and view all the answers

What is the correct result of multiplying the multinomials $(2x - 3y)(x^2 - 3xy)$?

$2x^3 - 9xy^2 + 5y$ Signup and view all the answers

When multiplying $(3m^3 + m^2 - 2m - 5)(m^2 - 5m - 6)$, what is the coefficient of $m^4$ in the final answer?

-14 Signup and view all the answers

What method is used to divide a monomial by another monomial?

Fundamental theorem of fractions or exponent rules Signup and view all the answers

Which term represents an error when expanding the expression $(3m^3 + m^2 - 2m - 5)(m^2 - 5m - 6)$?

$10m^2$ Signup and view all the answers

If you multiply the terms in $(2x)(3x^2)$, what is the resulting expression?

$6x^3$ Signup and view all the answers

What is the resulting expression of $(2x^2)(3x - 4)$ when multiplied?

$6x^3 - 8x^2$ Signup and view all the answers

How many terms are there in the final expanded expression of $(3m^3 + m^2 - 2m - 5)(m^2 - 5m - 6)$?

5 Signup and view all the answers

Which of the following is a common mistake when expanding the expression $(2x)(x^2 + 4x + 5)$?

$2x^3 + 4x^2 + 10x$ Signup and view all the answers

What is the value of $a^0$ for any non-zero real number $a$?

1 Signup and view all the answers

When simplifying the expression $6x^2y^3 - 12xy + 4xy^2$ divided by $3xy$, what will the first term of the result be?

$2x^2y^2$ Signup and view all the answers

Which step is NOT part of the process to divide a multinomial by a monomial?

Arrange the terms in ascending order Signup and view all the answers

If you divide $4x^3y - 8x^2y^2 + xy^3 + y^4$ by $2x - y$, what will the structure of the result contain?

Terms with descending powers of $x$ Signup and view all the answers

What is the result of simplifying $2x^0$?

$2$ Signup and view all the answers

In the expression $y^2 - 2x^2y + 4x^2y$, if you factor out $y$, what will be the resulting expression?

$y(4x^2 - 2x^2)$ Signup and view all the answers

What is a necessary condition for applying the zero exponent rule to a number $a$?

$a$ must be a non-zero number Signup and view all the answers

After performing the division of $6x^2y^3 - 12xy + 4xy^2$ by $3xy$, what type of terms will the result include?

Terms with variables raised to positive powers Signup and view all the answers

Which of the following expressions represents a valid operation involving division of a multinomial by a monomial?

$rac{5x^4 - 10x^2 + 15}{5x^2}$ Signup and view all the answers

Which operation is performed first when simplifying the expression $2y - 2x^2y + 4x^2y$?

Combine like terms Signup and view all the answers

What is the result of subtracting $1 5 w v$ from $1 2 w v$?

$-3wv$ Signup and view all the answers

What is the expression obtained when adding $3x + 2y - 4z$ to $(2x - 3y + 3z)$?

$5x - y - z$ Signup and view all the answers

What do you get when you subtract $5x^3 - 7x - 3$ from $3x + x^2 + 5$?

$-5x^3 + x^2 + 10$ Signup and view all the answers

What is the product of $(2x^2y^3)(-3yz^2)$?

$-6x^2y^3z^2$ Signup and view all the answers

When dividing $4x^2y^3(2xy^3 - 3xz^2)$ by $5$, what will be the leading term of the resulting expression?

$rac{8}{5}x^3y^3$ Signup and view all the answers

What is the result of adding $y + 5x^3 + 3y^3 - 10xy^2$?

$3y^3 + 5x^3 + y - 10xy^2$ Signup and view all the answers

What is the outcome when $4y - 2x$ is interchanged with $2x - 3y + 3z$?

$2x - 5y + 3z$ Signup and view all the answers

Study Notes

Constants and Variables

Constant: Represents a fixed value, can be a number or a letter.
Variable: A letter that can represent a set of real numbers, with values determined by the context.

Algebraic Expressions

Expressions consist of constants and/or variables combined through operations: addition, subtraction, multiplication, and division.
Term: Each monomial within an expression.
Binomial: An expression made up of two terms.
Multinomial: An expression composed of more than two terms.

Degree of Multinomial

Degree of a Term: The exponent of a literal factor within the term.
Example: In (3x^2y^3z), the degrees are (x = 2), (y = 3), (z = 1).

Powers and Exponents

Power: Combination of a base and an exponent.
Exponent: Indicates how many times the base acts as a factor.
Standard notation for the nth power: (b^n) is read as "b raised to the nth power."

Rules for Positive Integral Exponents

Rule 1: (a^m \cdot a^n = a^{m+n})
Rule 2: (a^m / a^n = a^{m-n}) (where (m > n) and (a \neq 0))
Rule 3: ((ab)^m = a^m b^m) (for (m > 0))
Rule 4: (\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}) (for (b \neq 0))
Rule 5: ((a^m)^n = a^{m \cdot n})

Multiplication of Algebraic Expressions

Use the distributive property for multiplying multinomials.
Example: ((2x - 3y)(x^2 - 3xy)) expands through distribution to combine like terms.

Division of Algebraic Expressions

Monomial by Monomial: Use rules of exponents and fundamental fraction theorem.
Multinomial by Monomial: Arrange terms by decreasing powers of a common factor; divide term by term.
Multinomial by Multinomial: Follow similar principles, applying polynomial long division when necessary.

Exercises

Various operations such as addition, subtraction, and multiplication of algebraic expressions require simplification according to the aforementioned rules and properties.

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Description

In this quiz, we explore the fundamentals of algebraic expressions, including constant and variable definitions, as well as operations involving integer exponents. Test your understanding of addition, subtraction, multiplication, and division of algebraic expressions covered in Lecture 1 of FNDMATH.