Algebra Class: Expressions and Identities

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Questions and Answers

What is the product of the monomials 4xy and 5x2y2?

The product is 20x3y3.

How does the commutative property apply to multiplication of monomials?

The commutative property allows us to rearrange the order of multiplication, leading to the same product regardless of the order.

Calculate the volume of a box with dimensions 2ax, 3by, and 5cz.

The volume is 30abcxyz.

What is the area of a rectangle with length 10m and breadth 5n?

<p>The area is 50mn.</p> Signup and view all the answers

Find the product of the monomials -4p and 7pq.

<p>The product is -28p^2q.</p> Signup and view all the answers

If the first monomial is 3x^2 and the second is 4xy, what is their product?

<p>The product is 12x^3y.</p> Signup and view all the answers

What happens to the product if one of the monomials is zero?

<p>The product becomes zero.</p> Signup and view all the answers

Complete the product of the monomials 2x and 2x.

<p>The product is 4x^2.</p> Signup and view all the answers

What is the sum of the expressions 7x2 - 4x + 5 and 9x - 10?

<p>The sum is 7x2 + 5x - 5.</p> Signup and view all the answers

When adding the expressions 7xy + 5yz - 3zx, 4yz + 9zx - 4y, and -2xy + 5x - 3zx, what is the final combined expression?

<p>The final expression is 5xy + 9yz + 3zx + 5x - 4y.</p> Signup and view all the answers

Explain how to perform subtraction when subtracting one algebraic expression from another.

<p>Subtraction is done by adding the additive inverse of the expression being subtracted.</p> Signup and view all the answers

What happens to terms in algebraic expressions if they do not have like terms during addition?

<p>They are carried through as they are.</p> Signup and view all the answers

What is the result of subtracting 5x2 - 4y2 + 6y - 3 from 7x2 - 4xy + 8y2 + 5x - 3y?

<p>The result is 2x2 - 4xy + 12y2 + 5x - 9y + 3.</p> Signup and view all the answers

In the expression 2x2 + 3x - 5, what are the like terms?

<p>The like terms are 2x2 and 3x.</p> Signup and view all the answers

If you have the expression 5 - 12, what is its equivalent addition operation?

<p>It is equivalent to 5 + (-12).</p> Signup and view all the answers

In the expression 4yz + 9zx - 4y, which term contributes to the y variable?

<p>-4y contributes to the y variable.</p> Signup and view all the answers

Using the distributive law, what is the product of $(4p^2 + 5p + 7) \times 3p$?

<p>The product is $12p^3 + 15p^2 + 21p$.</p> Signup and view all the answers

Simplify the expression $x(x - 3) + 2$ and evaluate it for $x = 1$.

<p>The simplified expression is $x^2 - 3x + 2$ and evaluates to $0$ when $x = 1$.</p> Signup and view all the answers

What is the result of $3y(2y - 7) - 3(y - 4) - 63$ when $y = -2$?

<p>The result is $21$.</p> Signup and view all the answers

Calculate the sum of $5m(3 - m)$ and $6m^2 - 13m$.

<p>The sum is $m^2 + 2m$.</p> Signup and view all the answers

Find the sum of $4y(3y^2 + 5y - 7)$ and $2(y^3 - 4y^2 + 5)$.

<p>The resulting expression is $14y^3 + 12y^2 - 28y + 10$.</p> Signup and view all the answers

What is the difference when subtracting $3pq(p - q)$ from $2pq(p + q)$?

<p>The difference is $-pq + 5pq^2$.</p> Signup and view all the answers

Using the distributive law, compute the expression $(12p^3 + 20p^2 - 28p) + (2p^3 - 8p^2 + 10)$.

<p>The simplified expression is $14p^3 + 12p^2 - 28p + 10$.</p> Signup and view all the answers

What is the expanded form of $2pq(p + q)$?

<p>The expanded form is $2p^2q + 2pq^2$.</p> Signup and view all the answers

What is the volume of a rectangular box with dimensions 5a, 3a^2, and 7a^4?

<p>The volume is $105a^7$.</p> Signup and view all the answers

Calculate the product of the following: 2, 4y, 8y^2, and 16y^3.

<p>The product is $1024y^6$.</p> Signup and view all the answers

Find the product of a, 2b, 3c, and 6abc.

<p>The product is $36a^2b^2c^2$.</p> Signup and view all the answers

Using the distributive law, what is the result of 3x × (5y + 2)?

<p>The result is $15xy + 6x$.</p> Signup and view all the answers

What is the product of xy, yz, and zx?

<p>The product is $xyz^2$.</p> Signup and view all the answers

What is the result of multiplying –3x by the binomial (–5y + 2)?

<p>The result is $15xy - 6x$.</p> Signup and view all the answers

Calculate the volume of the rectangular box with dimensions xy, 2x^2y, and 2xy^2.

<p>The volume is $4x^3y^4$.</p> Signup and view all the answers

What is the result of multiplying a monomial a^2 by the binomial (2ab - 5c)?

<p>The result is $2a^3b - 5a^2c$.</p> Signup and view all the answers

What is the sum of the expressions ab - bc, bc - ca, and ca - ab?

<p>The result is 0.</p> Signup and view all the answers

If we perform the operation a - b + ab, b - c + bc, and c - a + ac, what is the total expression?

<p>The sum is a + b + c + ab + bc + ac.</p> Signup and view all the answers

What is the result of 2pq - 3pq + 4 and 5 + 7pq - 3pq?

<p>The simplified result is 2pq + 9.</p> Signup and view all the answers

When subtracting 4a - 7ab + 3b + 12 from 12a - 9ab + 5b - 3, what is the expression you get?

<p>The result is 8a - 2ab + 2b - 15.</p> Signup and view all the answers

What expression do you get when you subtract 3xy + 5yz - 7zx from 5xy - 2yz - 2zx + 10xyz?

<p>The resulting expression is 2xy - 7yz + 5zx + 10xyz.</p> Signup and view all the answers

When subtracting 4p^2q - 3pq + 5pq^2 - 8p + 7q - 10 from 18 - 3p - 11q + 5pq - 2pq^2 + 5p^2q, what is the final outcome?

<p>The result is 8 - 2p + 4q - 10pq + 6p^2q.</p> Signup and view all the answers

Using the pattern of dots shown, how do you calculate the total number of dots produced by the expression m×n?

<p>You multiply m, the number of rows, by n, the number of columns.</p> Signup and view all the answers

What does the expression 4×9 represent in terms of algebraic multiplication?

<p>It represents the multiplication of 4 rows by 9 columns, resulting in a total of 36 dots.</p> Signup and view all the answers

What is the formula for the area of a rectangle and how does it change when the length is increased by 5 units and the breadth decreased by 3 units?

<p>The formula for the area of a rectangle is $l \times b$. If the length is increased by 5 units and the breadth decreased by 3 units, the new area is $(l + 5) \times (b - 3)$.</p> Signup and view all the answers

How would you express the total cost of bananas if the price per dozen is decreased by 2 and the quantity needed is decreased by 4 dozen?

<p>The total cost would be expressed as $(p - 2) \times (z - 4)$ where $p$ is the original price and $z$ is the original quantity in dozens.</p> Signup and view all the answers

In the context of speed and time, how would you calculate the distance traveled if speed is increased by a certain amount?

<p>Distance can be calculated using the formula $d = speed \times time$; if speed is increased, the new distance would be expressed as $(speed + increase) \times time$.</p> Signup and view all the answers

What would be the new volume of a rectangular box if its dimensions are increased by certain amounts?

<p>The new volume of a rectangular box can be expressed as $(l + length_{increase}) \times (b + breadth_{increase}) \times (h + height_{increase})$.</p> Signup and view all the answers

Describe a scenario involving interest calculation that requires multiplication of algebraic expressions.

<p>If the principal amount is $P$, the rate of interest is $R$, and time is $T$, the interest can be calculated using the formula $I = P \times R \times T$.</p> Signup and view all the answers

Can you provide a general expression for the area of a rectangle and how adjusting its dimensions affects the formula?

<p>The area of a rectangle is given by $A = l \times b$; changing the dimensions leads to $A = (l + x) \times (b + y)$ where $x$ and $y$ are the adjustments.</p> Signup and view all the answers

If the price of an item increases and you buy fewer quantities, how would that affect the total cost using algebraic expressions?

<p>The total cost would be represented as $(price + increase) \times (quantity - decrease)$, modifying both elements involved.</p> Signup and view all the answers

How can multiplication of algebraic expressions be used to model changes in a garden's area when planting?

<p>The area of a garden can be modeled as $A = length \times width$; if length increases and width decreases, the new area is $(length + increase) \times (width - decrease)$.</p> Signup and view all the answers

Flashcards

Adding Algebraic Expressions

Combining algebraic expressions with different terms by adding or subtracting their coefficients.

Subtracting Algebraic Expressions

Subtracting one algebraic expression from another by changing the signs of the terms in the expression being subtracted.

Multiplication of Monomials

The product of two or more monomials where the coefficients are multiplied and the variables are multiplied using the laws of exponents.

Multiplication of a Monomial and a Polynomial

The product of one monomial and one polynomial, where each term of the polynomial is multiplied by the monomial.

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Multiplication of Binomials

Multiplying two binomials by distributing each term of one binomial to each term of the other binomial.

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Multiplication of Polynomials

Multiplying two or more polynomials by systematically multiplying each term of one polynomial by all terms of the other polynomials.

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Squares of Binomials

A short form of writing the product of an expression multiplied by itself.

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Product of the Sum and Difference of Binomials

A short form of writing the product of the sum and difference of two expressions.

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Commutative Property of Multiplication

The order of multiplication does not affect the result. You can multiply monomials in any order.

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Multiplying Monomials

The product of monomials is found by multiplying their coefficients and adding their exponents for the same variables.

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Product of Monomials

The result of multiplying two or more monomials.

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Monomial

A single term consisting of a numerical coefficient and one or more variables raised to non-negative powers.

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Area of a Rectangle

The area of a rectangle is found by multiplying its length and breadth.

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Volume of a Rectangular Box

The volume of a rectangular box is found by multiplying its length, breadth, and height.

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Algebraic Expression

A combination of variables, constants, and operations (addition, subtraction, multiplication, division) without any equality sign.

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Algebraic Identity

A rule that simplifies a mathematical expression without changing its value.

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Trinomial

An algebraic expression with three terms.

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Polynomial

An expression with one or more terms, where the coefficients are non-zero and the exponents of variables are non-negative integers.

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Multiplying a Monomial by a Binomial

The process of multiplying a monomial by each term of a binomial.

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Multiplying a Monomial by a Trinomial

The process of multiplying a monomial by each term of a trinomial.

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Commutative Law

The property which states a × b = b × a, holds true for any numbers a and b.

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Distributive Law

The property that states a × (b + c) = (a × b) + (a × c), holds true for any numbers a, b, and c.

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Like Terms

Terms in an algebraic expression that have the same variable and exponent. Example: 3x² and 5x² are like terms.

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Combining Like Terms

The process of combining like terms in an expression to simplify it. Example: 3x + 5x = 8x

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Constant

A number that stands alone in an expression, without any variables. Example: 5 in the expression 3x + 5.

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Variable

A symbol that represents an unknown value. It can take on different values. Example: 'x' in the expression 2x + 3.

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Multiplication of algebraic expressions

The process of finding the product of two or more expressions involving variables.

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(l + 5) in the context of a rectangle

Representing a change in the length of a rectangle. The expression (l + 5) represents the original length (l) increased by 5 units.

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(b - 3) in the context of a rectangle

Representing a change in the breadth of a rectangle. The expression (b - 3) represents the original breadth (b) decreased by 3 units.

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Area of a rectangle with changed dimensions

The area of a rectangle is the product of its length and breadth. If the length is increased and the breadth is decreased, the new area is represented by multiplying the new length and breadth expressions.

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Cost of buying multiple items

The total cost of buying a certain number of items is found by multiplying the price per item by the number of items. Changes to the price or quantity affect the total cost.

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Distance covered

The distance covered by an object is found by multiplying its speed by the time taken. Changes to the speed or time impact the distance traveled.

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Simple interest calculation

The amount of simple interest earned or paid is found by multiplying the principal amount, the rate of interest, and the time period. Changes to any of these values will affect the interest.

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Simplifying Algebraic Expressions

Simplifying expressions involves using the distributive law to expand products and then combining like terms.

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Evaluating Algebraic Expressions

To evaluate an expression, substitute the given value for the variable and then simplify the expression.

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Multiplying Algebraic Expressions

Multiplication of algebraic expressions involves applying the distributive law to multiply each term of one expression by each term of the other.

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Study Notes

Algebraic Expressions and Identities

  • Algebraic expressions are combinations of variables, constants, and operations (addition, subtraction, multiplication, and division).
  • Examples of expressions include: x + 3, 2y – 5, 3x², 4xy + 7
  • Adding and subtracting algebraic expressions involves combining like terms.
  • Like terms have the same variables raised to the same powers.
  • Coefficients of like terms are added or subtracted.
  • Example: 7x² – 4x + 5 + 9x – 10 = 7x² + 5x – 5
  • Subtracting an expression is the same as adding the additive inverse of each term.
  • To subtract an expression, change the sign of each term in the expression and then add it to the other expression.
  • Example: (7x² – 4xy + 8y² + 5x – 3y) − (5x² - 4y² + 6y – 3) = 2x² – 4xy + 12y² + 5x – 9y + 3

Multiplication of Algebraic Expressions

  • The total number of dots in a pattern can be calculated by multiplying the number of rows by the number of columns.
  • The area of a rectangle is calculated by multiplying the length times the width (l × w).
  • The volume of a rectangular box is calculated by multiplying the length, width, and height (l × w × h).
  • Multiplying two or more algebraic expressions results in a new algebraic expression.
  • The process involves multiplying each term in one expression by every term in the other.
  • Example: 5x × 4x² = 20x³
  • Example: 4xy × 5x² y² × 6x³ y³= 120 x⁶ y⁶( combining like terms)

Multiplying a Monomial by a Monomial, Binomial, Trinomial

  • Expression with one term is a monomial
  • Expression with two terms is a binomial
  • Expression with three terms is a trinomial
  • Multiplying a monomial by a monomial, binomial, or trinomial uses the distributive property.
  • Each term of the monomial is multiplied by each term in the other expression.
  • Example: 3x (5y + 2) = 15xy + 6x
  • Example: 2x × ( 3x² + 2x + 1) = 6x³ + 4x² + 2x
  • In multiplying a polynomial by a polynomial, each term of one polynomial is multiplied by each term in the other and like terms are combined

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