Your Answer Should Be A Polynomial In Standard Form. ( 3 K + 4 ) ( 9 K + 5 ) = □ (3k + 4)(9k + 5) = \square ( 3 K + 4 ) ( 9 K + 5 ) = □

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Introduction

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In algebra, expanding and simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical expressions. In this article, we will focus on expanding and simplifying the product of two binomials, which is a fundamental concept in algebra. We will use the given expression $(3k + 4)(9k + 5) = \square$ as an example to demonstrate the steps involved in expanding and simplifying algebraic expressions.

What is a Binomial?

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A binomial is an algebraic expression consisting of two terms. It can be written in the form $ax + b$, where $a$ and $b$ are constants, and $x$ is a variable. In the given expression, $(3k + 4)$ and $(9k + 5)$ are binomials.

Expanding Binomials

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To expand a binomial, we need to multiply each term in the first binomial by each term in the second binomial. This process is called the distributive property. Let's expand the given expression using the distributive property.

Step 1: Multiply the First Term of the First Binomial by Each Term of the Second Binomial

We will multiply the first term of the first binomial, $3k$, by each term of the second binomial, $9k$ and $5$. This gives us:

$3k \cdot 9k = 27k^2$

$3k \cdot 5 = 15k$

Step 2: Multiply the Second Term of the First Binomial by Each Term of the Second Binomial

We will multiply the second term of the first binomial, $4$, by each term of the second binomial, $9k$ and $5$. This gives us:

$4 \cdot 9k = 36k$

$4 \cdot 5 = 20$

Step 3: Combine Like Terms

Now, we will combine like terms by adding or subtracting terms with the same variable and exponent. In this case, we have:

$27k^2 + 15k + 36k + 20$

Combining like terms, we get:

$27k^2 + 51k + 20$

Simplifying the Expression

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The expression $27k^2 + 51k + 20$ is the expanded form of the given expression $(3k + 4)(9k + 5)$. This is the standard form of a polynomial, where the terms are arranged in descending order of the exponent of the variable.

Conclusion

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In this article, we have demonstrated the steps involved in expanding and simplifying algebraic expressions. We have used the given expression $(3k + 4)(9k + 5) = \square$ as an example to show how to expand and simplify binomials. By following these steps, we can expand and simplify any algebraic expression, which is a crucial skill in solving equations and manipulating mathematical expressions.

Frequently Asked Questions

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Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that a single term can be multiplied by each term in a binomial.

Q: How do I expand a binomial?

A: To expand a binomial, you need to multiply each term in the first binomial by each term in the second binomial.

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is where the terms are arranged in descending order of the exponent of the variable.

Examples

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Example 1: Expand the expression $(2x + 3)(4x + 2)$

Using the distributive property, we get:

$2x \cdot 4x = 8x^2$

$2x \cdot 2 = 4x$

$3 \cdot 4x = 12x$

$3 \cdot 2 = 6$

Combining like terms, we get:

$8x^2 + 16x + 6$

Example 2: Expand the expression $(x + 2)(x + 5)$

Using the distributive property, we get:

$x \cdot x = x^2$

$x \cdot 5 = 5x$

$2 \cdot x = 2x$

$2 \cdot 5 = 10$

Combining like terms, we get:

$x^2 + 7x + 10$

Practice Problems

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Problem 1: Expand the expression $(3x + 2)(2x + 1)$

Problem 2: Expand the expression $(x + 4)(x + 3)$

Problem 3: Expand the expression $(2x + 5)(3x + 2)$

Solutions

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Solution 1: Expand the expression $(3x + 2)(2x + 1)$

Using the distributive property, we get:

$3x \cdot 2x = 6x^2$

$3x \cdot 1 = 3x$

$2 \cdot 2x = 4x$

$2 \cdot 1 = 2$

Combining like terms, we get:

$6x^2 + 7x + 2$

Solution 2: Expand the expression $(x + 4)(x + 3)$

Using the distributive property, we get:

$x \cdot x = x^2$

$x \cdot 3 = 3x$

$4 \cdot x = 4x$

$4 \cdot 3 = 12$

Combining like terms, we get:

$x^2 + 7x + 12$

Solution 3: Expand the expression $(2x + 5)(3x + 2)$

Using the distributive property, we get:

$2x \cdot 3x = 6x^2$

$2x \cdot 2 = 4x$

$5 \cdot 3x = 15x$

$5 \cdot 2 = 10$

Combining like terms, we get:

$6x^2 + 19x + 10$

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Introduction

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Algebraic expressions are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving equations and manipulating mathematical expressions. In this article, we will provide a comprehensive Q&A guide on algebraic expressions, covering topics such as expanding and simplifying expressions, binomials, and polynomials.

Q&A

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Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations.

Q: What is a binomial?

A: A binomial is an algebraic expression consisting of two terms. It can be written in the form $ax + b$, where $a$ and $b$ are constants, and $x$ is a variable.

Q: How do I expand a binomial?

A: To expand a binomial, you need to multiply each term in the first binomial by each term in the second binomial. This process is called the distributive property.

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is where the terms are arranged in descending order of the exponent of the variable.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to combine like terms by adding or subtracting terms with the same variable and exponent.

Q: What is a polynomial?

A: A polynomial is an algebraic expression consisting of one or more terms, where each term is a constant or a variable raised to a non-negative integer power.

Q: How do I add or subtract polynomials?

A: To add or subtract polynomials, you need to combine like terms by adding or subtracting the coefficients of the terms with the same variable and exponent.

Q: How do I multiply polynomials?

A: To multiply polynomials, you need to multiply each term in the first polynomial by each term in the second polynomial, and then combine like terms.

Q: What is the difference between a polynomial and a rational expression?

A: A polynomial is an algebraic expression consisting of one or more terms, where each term is a constant or a variable raised to a non-negative integer power. A rational expression is a fraction of two polynomials.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, and then cancel out any common factors.

Q: What is the difference between a polynomial and a trigonometric expression?

A: A polynomial is an algebraic expression consisting of one or more terms, where each term is a constant or a variable raised to a non-negative integer power. A trigonometric expression is an expression that involves trigonometric functions, such as sine, cosine, and tangent.

Q: How do I simplify a trigonometric expression?

A: To simplify a trigonometric expression, you need to use trigonometric identities and formulas to rewrite the expression in a simpler form.

Examples

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Example 1: Simplify the expression $(3x + 2)(2x + 1)$

Using the distributive property, we get:

$3x \cdot 2x = 6x^2$

$3x \cdot 1 = 3x$

$2 \cdot 2x = 4x$

$2 \cdot 1 = 2$

Combining like terms, we get:

$6x^2 + 7x + 2$

Example 2: Simplify the expression $(x + 4)(x + 3)$

Using the distributive property, we get:

$x \cdot x = x^2$

$x \cdot 3 = 3x$

$4 \cdot x = 4x$

$4 \cdot 3 = 12$

Combining like terms, we get:

$x^2 + 7x + 12$

Example 3: Simplify the expression $(2x + 5)(3x + 2)$

Using the distributive property, we get:

$2x \cdot 3x = 6x^2$

$2x \cdot 2 = 4x$

$5 \cdot 3x = 15x$

$5 \cdot 2 = 10$

Combining like terms, we get:

$6x^2 + 19x + 10$

Practice Problems

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Problem 1: Simplify the expression $(3x + 2)(2x + 1)$

Problem 2: Simplify the expression $(x + 4)(x + 3)$

Problem 3: Simplify the expression $(2x + 5)(3x + 2)$

Solutions

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Solution 1: Simplify the expression $(3x + 2)(2x + 1)$

Using the distributive property, we get:

$3x \cdot 2x = 6x^2$

$3x \cdot 1 = 3x$

$2 \cdot 2x = 4x$

$2 \cdot 1 = 2$

Combining like terms, we get:

$6x^2 + 7x + 2$

Solution 2: Simplify the expression $(x + 4)(x + 3)$

Using the distributive property, we get:

$x \cdot x = x^2$

$x \cdot 3 = 3x$

$4 \cdot x = 4x$

$4 \cdot 3 = 12$

Combining like terms, we get:

$x^2 + 7x + 12$

Solution 3: Simplify the expression $(2x + 5)(3x + 2)$

Using the distributive property, we get:

$2x \cdot 3x = 6x^2$

$2x \cdot 2 = 4x$

$5 \cdot 3x = 15x$

$5 \cdot 2 = 10$

Combining like terms, we get:

$6x^2 + 19x + 10$

Conclusion

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In this article, we have provided a comprehensive Q&A guide on algebraic expressions, covering topics such as expanding and simplifying expressions, binomials, and polynomials. We have also provided examples and practice problems to help you understand and apply the concepts. By following the steps and examples provided in this article, you should be able to simplify and expand algebraic expressions with confidence.