Math 8 - Factoring Polynomials | Algebra

 Factoring Polynomials

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Algebraic factoring plays a significant role in industrial development in various ways. Algebraic factoring is not only a fundamental concept in mathematics but also a crucial tool for industrial development. Its applications span across various industries, aiding in problem-solving, decision-making, optimization, and innovation. The use of Algebra are mostly common in the following list below.

1. Efficient Problem Solving: 

   Algebraic factoring helps engineers and scientists solve complex equations and models more efficiently. In industries such as manufacturing, logistics, and engineering, efficient problem-solving leads to better designs, optimization of processes, and cost savings.

2. Optimization: 

   Many industrial processes involve maximizing or minimizing certain factors, such as minimizing production costs or maximizing product quality. Factoring allows for the simplification and manipulation of equations, making it easier to identify optimal solutions.

3. Financial Analysis: 

   Industries heavily rely on financial analysis, where algebraic factoring is essential for understanding and optimizing financial models, risk assessments, and investment decisions.

4. Engineering and Design: 

   Algebraic factoring is used in designing and analyzing systems, structures, and machinery. It helps engineers determine critical points, stability, and performance characteristics of products and systems.

5. Quality Control and Assurance: 

   Algebraic factoring aids in statistical analysis, which is vital for quality control and assurance in manufacturing and production processes. It helps identify patterns and trends in data, leading to better decision-making.

6. Modeling and Simulation: 

   Many industries use algebraic factoring to create mathematical models and simulations. This is crucial in fields like aerospace, automotive, and electronics for predicting behavior, performance, and potential challenges.

7. Resource Allocation: 

   Industries often need to allocate resources efficiently. Algebraic factoring assists in formulating and solving resource allocation problems, leading to optimized utilization of materials, labor, and energy.

8. Predictive Analysis: 

   Algebraic factoring is a foundation for predictive modeling and analysis. Industries can use these models to forecast trends, demand, and market behavior, aiding in strategic planning.

9. Process Improvement: 

   Algebraic factoring helps identify relationships between variables in industrial processes. By understanding these relationships, industries can implement targeted process improvements to enhance efficiency and productivity.

10. Innovation and Research: 

    Algebraic factoring is essential in research and development, enabling scientists and innovators to manipulate equations, test hypotheses, and make advancements in various fields.

Factoring polynomials is the process of expressing a polynomial as a product of simpler polynomials or monomials. It involves breaking down a polynomial into its constituent factors, which makes it easier to work with and can provide insights into the polynomial's behavior, roots, and other properties.

The main goal of factoring is to find the polynomial's prime factors or irreducible factors. Factoring can be particularly useful for various mathematical operations, such as simplifying expressions, solving equations, and graphing functions.

There are several methods for factoring polynomials, including:

1. Common Factor (Greatest Common Factor - GCF) Factoring: If the polynomial has a common factor among all its terms, you can factor it out. This helps simplify the polynomial and reveal its common factor. 

   Example: Factoring out the GCF in the polynomial yields  .

2. Factoring by Grouping: This method is applicable when the polynomial has four or more terms, and terms can be grouped in pairs that have common factors. Example: Factoring by grouping in the polynomial 

   
   

   which can be further factored as

   .

3. Difference of Squares Factoring: This method is used when the polynomial can be expressed as the difference of two perfect squares. Example: Factoring the polynomial,

   

4. Trinomial Factoring (Quadratic Factoring): This method involves factoring a quadratic polynomial (trinomial) into the product of two binomials. Example: Factoring the quadratic polynomial,
   

    
   

5. Special Factoring Patterns: Certain polynomial forms have specific factoring patterns, such as perfect squares, the sum or difference of cubes, etc. 

   Example: Factoring the polynomial,

    

6. Trial and Error: For more complex polynomials, trial and error can sometimes be used to identify potential factors.

Here's how you can factor a binomial, or even polynomial, with a common monomial factor:

1.    Identify the Common Factor: Look for a factor that is present in all the terms of the polynomial. This factor can be a constant or a variable raised to a certain power.

2.    Factor Out the Common Factor: Divide each term of the polynomial by the common factor. This step involves dividing each coefficient and variable term by the common factor.

3.    Write the Factored Form: The factored polynomial will have the common factor outside of a set of parentheses containing the quotient of each term divided by the common factor.

      

    1. Common Factor (Greatest Common Factor - GCF) Factoring

     The Common Factor (Greatest Common Factor - GCF) Factoring is a method used in algebra to simplify a polynomial expression by factoring out the greatest common factor (GCF) of its terms. The GCF is the largest expression that divides evenly into all the terms of the polynomial. Factoring out the GCF helps to simplify the expression and reveal a common factor shared by all terms.

     

      Illustration :

     

     Given the expression 2x+4, notice that there's "2" in 2x and there's also "2" in 4.

     Therefore, "2" is the common factor of 2x + 4. Hence, 2x+2*2 is the factor-ready form of the said expression.

     Thus, .

     

     As you notice that you must have at least one "common" factor so you can apply the monomial factoring in a given expression.

     Here's another example.

     Given the expression ax + bx.  What is the obvious "common factor" in this expression? the common factor is "x", as you can see there's x in "ax" and there's also x in "bx". We can actually take that "x" off from the expression and put it outside of the parenthesis where the inside is the terms that have no common factors.   

Let's Practice-1: 

Common Binomial Factoring

 From the given expressions, find the common factor and factor it out.

    

    

   

    
  

Let's Practice-2: 

Common Factoring from Trinomial Expression

A Trinomial expression is a mathematical expression that consists of three terms connected by addition or subtraction operators. In general, a Trinomial expression can be written in the form:

From the given expressions, find the common factor and factor it out.

   

   

    

 

 

 

 

Let's Practice-3: 

Common Factoring Binomials and Trinomials Expresions.

2. Factoring by Grouping

Factoring by grouping is a technique in algebra used to factor a polynomial with four or more terms. It involves grouping the terms in pairs, finding the greatest common factor (GCF) for each pair, factoring out the GCF from each pair, and then looking for a common factor that can be factored out from the resulting terms. This method helps simplify the polynomial and make it easier to factor further or solve equations.

Let's Practice-1:

Example 1: $\mathrm{ }$

Factor: 

 

 

$Answer:$

 

$$

Example 2:: $\mathrm{ }$

Factor:

 

 

 

$\mathrm{Answer:}$

$\mathrm{ }$

 

Example 3::  $\mathrm{<br>}$

Factor: 

 

$\mathrm{}$

Answer: $$

$$

$<br>$

$$

$\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><u><span\; style="color:\; \#2b00fe;\; font-size:\; xx-small;"><span\; class="mspace"\; style="margin-right:\; 0.2222em;"></span></span></u><span\; class="mbin"></span><span\; class="mspace"\; style="margin-right:\; 0.2222em;"></span></span></span></span></span>}$

Example 4::  $\mathrm{<br>}$

Factor:  

  

 

Answer:

 

 

$$

 

Example 5:

Factor$\mathrm{ \; <br>}$

Answer:

$$

$$$<h3\; style="text-align:\; left;"><b>3.\; </b><b>Difference\; of\; Squares\; Factoring</b></h3>$

$$$Difference\; of\; squares\; factoring\; is\; a\; technique\; in\; algebra\; used\; to\; factor\; a\; polynomial\; that\; can$

$be\; written\; as\; the\; difference\; between\; two\; perfect\; squares.\; A\; perfect\; square\; is\; a\; number\; or\; expression$

$that\; is\; the\; result\; of\; squaring\; a\; whole\; number.\; The\; difference\; of\; squares\; factoring\; involves$

$breaking\; down\; the\; polynomial\; into\; two\; square\; terms\; that\; are\; subtracted\; from\; each\; other.$

$$

$<p>The\; pattern\; for\; the\; difference\; of\; squares\; factoring\; is</p>$

$<p\; style="text-align:\; left;"> <img\; alt="equation"\; id="output"\; src="https://latex.codecogs.com/svg.image?a^\{2\}-b^\{2\}=(a+b)(a-b)"></p>$

$<p>Where:</p>$$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi> </mi></mrow></semantics></math></span></span></span>$

$a$ and $\mathrm{<i>b</i> }$are terms that represent perfect squares.

$\; <br>$

$<b><span\; style="color:\; \#990000;">Example\; 1:</span></b>$

$<b><span\; style="color:\; \#990000;"> </span></b>$

$$

Factor:$$

$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn> </mn></mrow></semantics></math></span></span></span>$

Answer:$$

$$

$$

$$

$$$<b><span\; style="color:\; \#990000;">\; <br></span></b>$

$<b><span\; style="color:\; \#990000;">Example\; 2:</span></b>$

$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn> </mn></mrow></semantics></math></span></span></span>$

Factor:  $<math\; xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn><b></b></mn></mrow></semantics></math>$$<math\; xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn><b> </b></mn></mrow></semantics></math>$

$<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn> </mn></mrow></semantics></math></span></span></span>$

Answer:$<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="strut"\; style="height:\; 1em;\; vertical-align:\; -0.25em;"></span></span></span></span></span>$

$$

$<b><span\; style="color:\; \#990000;"> <br></span></b>$

$<b><span\; style="color:\; \#990000;">Example\; 3:</span></b>$

$$

Factor: ${}^{}$

Answer:

$$

$<b><span\; style="color:\; \#990000;">Example\; 4:</span></b>\; <br>$

$$

Factor:  

Answer:

$$ 

$$Example 5:

Factor:

Answer:  $$

$$

$$4. Trinomial Factoring (Quadratic Factoring)$$

Trinomial factoring, also known as quadratic factoring, is a method in algebra used to factorize

a quadratic polynomial with three terms. The goal is to express the quadratic trinomial as the product of two binomials. These binomials represent two separate factors that, when multiplied together, give you the original quadratic expression.

 

 The general form of a quadratic trinomial is: $$

$$

$$$\mathrm{<img\; alt="equation"\; height="17"\; id="output"\; src="https://latex.codecogs.com/svg.image?\backslash inline\; \backslash small\; ax^\{2\}+bx+cax^\{2\}+bx+c"\; width="200">}$

$$

$$

$$The process involves finding two numbers (let's call them $m$m and $n$) that, when multiplied, give you the product of $a$ and $c$, and when added, give you $b$. Then, you use these numbers to rewrite the middle-term 

($bx$) of the trinomial, and finally, factor by grouping or applying other methods to factor out the common factors.

$$

$$

$$$Let\text{'}s\; say\; we\; have\; the\; quadratic\; trinomial:$

$$$\mathrm{ }$

$$$\mathrm{ }$$\mathrm{<img\; alt="equation"\; height="17"\; id="output"\; src="https://latex.codecogs.com/svg.image?\backslash inline\; \backslash small\; 2x^\{2\}+7x+32x^\{2\}+7x+3"\; width="200">}$

Let's Practice:

Example 1:

Given: ${}^{}$

Answer: $(x+2)(x+3)$

Example 2:

Given:

 

Answer: $(2x+3)(x-5)$

Example 3:

Given: $\mathrm{}$

$\mathrm{ }$

Answer: $(3x+3)(x+1)$ or simply $3(x+1{)}^{}$

Example 4:

 

Given: 

 

Answer:

 

 

Example 5:

 

Given:  

$\mathrm{ }$

Answer:  

or $\mathrm{5(}$$x+1$$\mathrm{)}(x+1{)}^{}$

 

5-Special Factoring Patterns

Special factoring patterns are specific methods or rules that can be used to simplify and factor certain types of algebraic expressions. These patterns help you quickly recognize and break down expressions into simpler forms. They are like shortcuts for factoring, making it easier to work with and understand more complex equations.

Example 1: Perfect Square Trinomial:

 

Given the Expression:  

Factored Form:

${\mathrm{ }}^{}$

${\mathrm{ }}^{}$

${\mathrm{<span\; style="color:\; \#990000;"><b>Example\; 2: \; </b></span><span\; style="color:\; \#990000;"></span>Difference\; of\; Squares:}}^{}$${\mathrm{<br>}}^{}$

${\mathrm{<span\; style="color:\; \#990000;"><b> </b></span>}}^{}$

${\mathrm{<span>Given\; the\; Expression:</span><span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi> </mi></msup></mrow></semantics></math></span></span></span>}}^{}$${\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; class="katex-mathml"><math\; xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>\; </mi></msup></mrow></semantics></math><br></span></span></span>}}^{}$

${\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mspace"\; style="margin-right:\; 0.2222em;"></span></span><span\; class="base"><span\; class="strut"\; style="height:\; 0.6444em;"></span><span\; class="mord"></span></span></span></span></span>}}^{}$

${\mathrm{Factored\; Form:}}^{}$${\mathrm{ }}^{}$

${\mathrm{ }}^{}$

${\mathrm{<span\; style="color:\; \#990000;"><b> </b></span>}}^{}$

${\mathrm{<span\; style="color:\; \#990000;"><b>Example\; 3: \; </b></span>}}^{}$${\mathrm{Difference\; of\; Cubes:}}^{}$

 

Given the Expression:  

$\mathrm{ }$

Factored Form: 

 

 

Example 4: Sum of Two Cubes

Given the Expression:  

 

Factored Form: 

 

 

Example 5: Quadratic Trinomial (AC Method):

 

Given the Expression:   

 

Factored Form:

$\; <br>$

$$$<h3\; style="text-align:\; left;"><span\; style="font-family:\; arial;"><b>6-Trial\; and\; Error\; Factoring</b></span></h3>$

$$$Trial\; and\; error\; factoring\; is\; a\; method\; used\; in\; algebra\; to\; find\; the\; factors\; of\; an\; expression$

$by\; trying\; different\; combinations\; and\; seeing\; which\; ones\; work$$.\; It\text{'}s\; like\; experimenting\; with$

$numbers\; to\; break\; down\; a\; math\; problem\; into\; simpler\; parts.\; You\; test\; different\; options\; until$

$you\; find\; the\; right\; combination\; that\; helps\; you\; rewrite\; the\; expression\; as\; a\; product\; of\; simpler$

$terms.\; It\; might\; take\; some\; attempts,\; but\; it\text{'}s\; a\; way\; to\; figure\; out\; how\; to\; factorize\; an\; equation.$

  

Example 1. Trial and Error with a Quadratic Trinomial: 

 

Expression: 

Trial and Error Factors:    

Example 3. Trial and Error with a Difference of Squares: 

Expression:

Trial and Error Factors:   

 

Example 4. Trial and Error with a Sum of Cubes: 

Expression:  

Trial and Error Factors: 

Example 5. Trial and Error with a Complex Trinomial: 

Expression:

Trial and Error Factors:

$$

$Factoring\; Exercises\; with\; Answers.$

Greatest Common Factor (GCF)

 

 

 

 

 

 

 

$Factoring\; by\; Grouping$

 

 

$\mathrm{ }$

$\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"> </span></span></span></span></span>}$

 

 

Difference of Squares Factoring

 

1.)  $x^2-4$

 

$2.)$$16y^2-\mathrm{25\; =}$

$\mathrm{ }$

$\mathrm{3.) }$$9a^2-1$

 

$4.)$$25x^2-36$

 

$5.)$$49b^2-9$

 

$6.)$$81x^2-100$

 

$7.)$$36a^2-81$

$8.)$$64x^2-49$

 

$9.)$$144y^2-\mathrm{25\; =}$

$\mathrm{ }$

$\mathrm{10.) }$$\mathrm{ }$

 

Trinomial Factoring

 

$1.)$ $x^2+5x+\mathrm{6\; =}$

$\mathrm{ }$

$\mathrm{2.) }$$2x^2-7x-3$

 

$3.)$$3x^2+8x+4$

 

$4.)$$4x^2-12x+9$

 

$5.)$$x^2-9x+18$

 

$6.)$$6x^2+11x-\mathrm{10\; = }$

$\mathrm{ }$

$\mathrm{7.) }$

$\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"> </span></span></span></span></span>}$

$\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"> </span></span></span></span></span>}$

$\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"></span></span></span></span>}$

$\mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mspace"\; style="margin-right:\; 0.2222em;">10.)\; </span></span><span\; class="base"><span\; class="strut"\; style="height:\; 0.6444em;"></span><span\; class="mord"></span></span></span></span></span>}$$$$6x^2-11x-10$

$$ 

$$Special Factoring Patterns (this also applies to Trial and Error factoring)

$$

$$

$$

$$1.) Difference of Squares: 

$$ 

$Expression:$$x^2-\mathrm{9 }$

$$ 

$$ 

$$ 2.) Perfect Square Trinomial: 

$$ 

$Expression:$$x^2+6x+9$

$$ 

$$ 

$$3.) Sum of Cubes: 

$$ 

$Expression:$$x^3+8$

$$ 

$$ 

$$

$$4.) Difference of Cubes: 

$$ 

$Expression:$$8x^3-\mathrm{27=}$

$$$\mathrm{ }$

$$$\mathrm{ }$

$$ \mathrm{<b>5.)\; Quadratic\; Trinomial\; (AC\; Method): </b>}$$

$$ \mathrm{<b> </b>}$$

$\mathrm{}$

$$$\mathrm{<br>}$

$$$\mathrm{<br>}$

$$ \mathrm{<b>6.)\; Perfect\; Square\; Trinomial:</b> }$$

$$ \mathrm{ }$$

$$ \mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"><span\; class="mord\; mathnormal"> </span></span></span></span></span></span>}$$

$$ \mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"><span\; class="mord\; mathnormal"> </span></span></span></span></span></span>}$$

$$ \mathrm{<b>7.)\; Sum\; of\; Cubes:</b> }$$

$$ \mathrm{ }$$

$$ \mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"><span\; class="mord\; mathnormal"><br></span></span></span></span></span></span>}$$

$$ \mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"><span\; class="mord\; mathnormal"><br></span></span></span></span></span></span>}$$

$$ \mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"><span\; class="mord\; mathnormal"></span></span></span></span></span></span><b>8.)\; Quadratic\; Trinomial\; (AC\; Method):</b> }$$

$$ \mathrm{ }$$

$$ \mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"> </span></span></span></span></span>}$$

$$ \mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"> \; <br></span></span></span></span></span>}$$

$$ \mathrm{<b>9.)\; Perfect\; Square\; Trinomial:</b> }$$

$$ \mathrm{ }$$

$$ \mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"><br></span></span></span></span></span>}$$

$$ \mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"><br></span></span></span></span></span>}$$

$$ \mathrm{<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"></span></span></span></span></span><b>10.)\; Sum\; of\; Cubes:</b> }$$

$$ \mathrm{ }$$

$Answers\; To\; Odd\; Number\; Problems.$

$GCF$

 

 

9.)  $8a(2a-b^2)$
 

$Factoring\; by\; Grouping$

1.)  $(x+2)(2x^2+3)$

3.)  $(x-2)(5xy+3)$

$$

$5.)\;$$$ (2x-3)(3x^2+1)$$

$$ $$

$$ <span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mspace"\; style="margin-right:\; 0.2222em;"></span></span></span></span></span>$$

$9.)$$(3x-5)(3xy+2)$

 

Difference of Squares

 

1.)  $(x+2)(x-2)$

 

$3.)$$(3a+1)(3a-1)$

 

$5.)$$(7b+3)(7b-3)$

 

$7.)$$(6a+9)(6a-9)$

9.)  $(12y+5)(12y-5)$

 

Trinomial Factoring

1.)  $(x+2)(x+3)$

$$

$3.)\;$

$<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mspace"\; style="margin-right:\; 0.2222em;"></span></span><span\; class="base"><span\; class="strut"\; style="height:\; 1em;\; vertical-align:\; -0.25em;"></span><span\; class="mord"></span><span\; class="mspace"\; style="margin-right:\; 0.2222em;"></span><span\; class="mbin"></span></span></span></span></span>$

$<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mspace"\; style="margin-right:\; 0.2222em;">5.) \; </span></span><span\; class="base"><span\; class="strut"\; style="height:\; 1em;\; vertical-align:\; -0.25em;"></span><span\; class="mord"></span></span></span></span></span>$$(x-6)(x-3)$

 

$7.)$$(5x-3)(x-1)$

 

$9.)$ $(2x+1)(x+3)$

Special Factoring Patterns

 

1.)  $(x+3)(x-3)$

 

$3.)$ $(x+2)(x^2-2x+4)$ 

 

5.)  $(2x+1)(x+3)$

$$

$7.)\;$

$<span\; class="math\; math-inline"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mclose"> </span></span></span></span></span>$

 

 

Other References:

 

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