Simplify: ${\left(15 P^8 Q^9 - 30 P^7 Q^{10} + 25 P^6 Q^5\right) \div \left(5 P^3 Q^4\right)}$13. Simplify: ${\left(-6 S^5 T^9 - 54 S^4 T^8 + 18 S^3 T^6 - 24 S^2 T^3\right) \div \left(-6 S^3 T^3\right)}$14. Simplify:

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill to master. In this article, we will explore the process of simplifying algebraic expressions, focusing on the division of polynomials. We will use real-world examples to illustrate the concepts and provide step-by-step solutions to three problems.

What is Simplification in Algebra?

Simplification in algebra refers to the process of reducing a complex expression to its simplest form. This involves combining like terms, canceling out common factors, and rearranging the terms to make the expression more manageable. Simplification is an essential skill in algebra, as it helps to:

  • Reduce the complexity of an expression
  • Make it easier to solve equations and inequalities
  • Identify patterns and relationships between variables
  • Improve the accuracy and efficiency of calculations

Steps to Simplify Algebraic Expressions

Simplifying algebraic expressions involves several steps:

  1. Identify like terms: Like terms are terms that have the same variable(s) raised to the same power. For example, 2x and 3x are like terms.
  2. Combine like terms: Combine like terms by adding or subtracting their coefficients. For example, 2x + 3x = 5x.
  3. Cancel out common factors: Cancel out common factors by dividing both the numerator and denominator by the common factor. For example, (6x^2 + 12x) / (2x) = 3x + 6.
  4. Rearrange the terms: Rearrange the terms to make the expression more manageable. For example, (x + 2) / (x - 1) can be rearranged to (x - 1 + 3) / (x - 1).

Problem 1: Simplify \[(15p8q9−30p7q10+25p6q5)÷(5p3q4)\[\left(15 p^8 q^9 - 30 p^7 q^{10} + 25 p^6 q^5\right) \div \left(5 p^3 q^4\right)

To simplify this expression, we need to follow the steps outlined above.

Step 1: Identify like terms

The given expression has three terms: 15p8q9, -30p7q10, and 25p6q5. We can identify like terms by looking at the variables and their exponents.

Step 2: Combine like terms

We can combine like terms by adding or subtracting their coefficients. In this case, we have:

  • 15p8q9
  • -30p7q10
  • 25p6q5

We can combine the first two terms by subtracting their coefficients:

15p8q9 - 30p7q10 = -15p7q10 + 15p8q9

Step 3: Cancel out common factors

We can cancel out common factors by dividing both the numerator and denominator by the common factor. In this case, we have:

(15p8q9 - 30p7q10 + 25p6q5) / (5p3q4)

We can cancel out the common factor of 5 by dividing both the numerator and denominator by 5:

(3p8q9 - 6p7q10 + 5p6q5) / (p3q4)

Step 4: Rearrange the terms

We can rearrange the terms to make the expression more manageable. In this case, we have:

(3p8q9 - 6p7q10 + 5p6q5) / (p3q4)

We can rearrange the terms by grouping the like terms together:

(3p8q9 - 6p7q10) / (p3q4) + (5p6q5) / (p3q4)

We can simplify the expression further by canceling out the common factor of p3q4:

3p5q5 - 6p4q6 + 5p3q1

The final answer is: 3p5q5 - 6p4q6 + 5p3q1

Problem 2: Simplify \[(−6s5t9−54s4t8+18s3t6−24s2t3)÷(−6s3t3)\[\left(-6 s^5 t^9 - 54 s^4 t^8 + 18 s^3 t^6 - 24 s^2 t^3\right) \div \left(-6 s^3 t^3\right)

To simplify this expression, we need to follow the steps outlined above.

Step 1: Identify like terms

The given expression has four terms: -6s5t9, -54s4t8, 18s3t6, and -24s2t3. We can identify like terms by looking at the variables and their exponents.

Step 2: Combine like terms

We can combine like terms by adding or subtracting their coefficients. In this case, we have:

  • -6s5t9
  • -54s4t8
  • 18s3t6
  • -24s2t3

We can combine the first two terms by subtracting their coefficients:

-6s5t9 - 54s4t8 = -54s4t8 - 6s5t9

We can combine the last two terms by subtracting their coefficients:

18s3t6 - 24s2t3 = 18s3t6 - 24s2t3

Step 3: Cancel out common factors

We can cancel out common factors by dividing both the numerator and denominator by the common factor. In this case, we have:

(-6s5t9 - 54s4t8 + 18s3t6 - 24s2t3) / (-6s3t3)

We can cancel out the common factor of -6 by dividing both the numerator and denominator by -6:

(s5t9 + 9s4t8 - 3s3t6 + 4s2t3) / (s3t3)

Step 4: Rearrange the terms

We can rearrange the terms to make the expression more manageable. In this case, we have:

(s5t9 + 9s4t8 - 3s3t6 + 4s2t3) / (s3t3)

We can rearrange the terms by grouping the like terms together:

(s5t9 + 9s4t8) / (s3t3) - (3s3t6 + 4s2t3) / (s3t3)

We can simplify the expression further by canceling out the common factor of s3t3:

s2t6 + 9s1t5 - 3s0t3 - 4s(-1)t0

The final answer is: s2t6 + 9s1t5 - 3s0t3 - 4s(-1)t0

Problem 3: Simplify \[(2x3+5x2−3x+1)÷(x+2)\[\left(2x^3 + 5x^2 - 3x + 1\right) \div \left(x + 2\right)

To simplify this expression, we need to follow the steps outlined above.

Step 1: Identify like terms

The given expression has four terms: 2x^3, 5x^2, -3x, and 1. We can identify like terms by looking at the variables and their exponents.

Step 2: Combine like terms

We can combine like terms by adding or subtracting their coefficients. In this case, we have:

  • 2x^3
  • 5x^2
  • -3x
  • 1

We can combine the first two terms by adding their coefficients:

2x^3 + 5x^2 = 5x^2 + 2x^3

We can combine the last two terms by subtracting their coefficients:

-3x + 1 = 1 - 3x

Step 3: Cancel out common factors

We can cancel out common factors by dividing both the numerator and denominator by the common factor. In this case, we have:

(2x^3 + 5x^2 - 3x + 1) / (x + 2)

We can cancel out the common factor of x + 2 by dividing both the numerator and denominator by x + 2:

(2x^3 + 5x^2 - 3x + 1) / (x + 2)

We can simplify the expression further by canceling out the common factor of x + 2:

2x^2 + 3x - 1

The final answer is

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Introduction

Simplifying algebraic expressions is a fundamental concept in mathematics, and it can be a challenging task for many students. In this article, we will provide a Q&A guide to help you understand the process of simplifying algebraic expressions.

Q: What is simplification in algebra?

A: Simplification in algebra refers to the process of reducing a complex expression to its simplest form. This involves combining like terms, canceling out common factors, and rearranging the terms to make the expression more manageable.

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, 2x and 3x are like terms.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract their coefficients. For example, 2x + 3x = 5x.

Q: What are common factors?

A: Common factors are factors that are present in both the numerator and denominator of an expression. For example, in the expression (6x^2 + 12x) / (2x), the common factor is 2x.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to divide both the numerator and denominator by the common factor. For example, (6x^2 + 12x) / (2x) = 3x + 6.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Not combining like terms
  • Not canceling out common factors
  • Not rearranging the terms to make the expression more manageable
  • Not checking for errors in the calculation

Q: How do I check my work when simplifying algebraic expressions?

A: To check your work when simplifying algebraic expressions, you need to:

  • Read the expression carefully to make sure you understand what it means
  • Identify the like terms and combine them
  • Cancel out common factors
  • Rearrange the terms to make the expression more manageable
  • Check for errors in the calculation

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

  • Solving equations and inequalities
  • Graphing functions
  • Modeling real-world situations
  • Making predictions and forecasts

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

  • Working on problems and exercises
  • Using online resources and tools
  • Asking for help from a teacher or tutor
  • Joining a study group or math club

Q: What are some common algebraic expressions that require simplification?

A: Some common algebraic expressions that require simplification include:

  • Fractions and decimals
  • Exponents and roots
  • Polynomials and rational expressions
  • Trigonometric expressions

Q: How can I simplify a fraction?

A: To simplify a fraction, you need to:

  • Identify the numerator and denominator
  • Find the greatest common factor (GCF) of the numerator and denominator
  • Divide both the numerator and denominator by the GCF

Q: How can I simplify an exponent?

A: To simplify an exponent, you need to:

  • Identify the base and exponent
  • Apply the rules of exponents (e.g. a^m * a^n = a^(m+n))
  • Simplify the expression

Q: How can I simplify a polynomial?

A: To simplify a polynomial, you need to:

  • Identify the terms and coefficients
  • Combine like terms
  • Cancel out common factors
  • Rearrange the terms to make the expression more manageable

Q: How can I simplify a rational expression?

A: To simplify a rational expression, you need to:

  • Identify the numerator and denominator
  • Find the greatest common factor (GCF) of the numerator and denominator
  • Divide both the numerator and denominator by the GCF
  • Simplify the expression

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Not combining like terms
  • Not canceling out common factors
  • Not rearranging the terms to make the expression more manageable
  • Not checking for errors in the calculation

Q: How can I check my work when simplifying algebraic expressions?

A: To check your work when simplifying algebraic expressions, you need to:

  • Read the expression carefully to make sure you understand what it means
  • Identify the like terms and combine them
  • Cancel out common factors
  • Rearrange the terms to make the expression more manageable
  • Check for errors in the calculation

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

  • Solving equations and inequalities
  • Graphing functions
  • Modeling real-world situations
  • Making predictions and forecasts

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

  • Working on problems and exercises
  • Using online resources and tools
  • Asking for help from a teacher or tutor
  • Joining a study group or math club

Q: What are some common algebraic expressions that require simplification?

A: Some common algebraic expressions that require simplification include:

  • Fractions and decimals
  • Exponents and roots
  • Polynomials and rational expressions
  • Trigonometric expressions

Conclusion

Simplifying algebraic expressions is a fundamental concept in mathematics, and it requires practice and patience to master. By following the steps outlined in this article, you can simplify algebraic expressions and apply them to real-world problems. Remember to combine like terms, cancel out common factors, and rearrange the terms to make the expression more manageable. With practice and dedication, you can become proficient in simplifying algebraic expressions and apply them to a wide range of mathematical and real-world problems.