Select The Correct Answer From Each Drop-down Menu.What Is The Factored Form Of This Expression? 27 T 3 − 36 T 2 − 12 T + 16 = ( □ ) ( □ ∨ ) ( □ ∨ 27t^3 - 36t^2 - 12t + 16 = (\square)(\square \vee)(\square \vee 27 T 3 − 36 T 2 − 12 T + 16 = ( □ ) ( □ ∨ ) ( □ ∨ ]

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Introduction

Factoring expressions is a fundamental concept in algebra that involves breaking down an expression into simpler components. It is a crucial skill for solving equations, graphing functions, and simplifying complex expressions. In this article, we will explore the factored form of a given expression and provide a step-by-step guide on how to factor it.

What is Factoring?

Factoring is the process of expressing an expression as a product of simpler expressions, called factors. These factors can be numbers, variables, or a combination of both. Factoring an expression can help us simplify it, identify its roots, and solve equations.

The Given Expression

The given expression is:

27t3−36t2−12t+1627t^3 - 36t^2 - 12t + 16

Our goal is to factor this expression into the form:

(□)(□∨)(□∨)(\square)(\square \vee)(\square \vee)

Step 1: Factor Out the Greatest Common Factor (GCF)

The first step in factoring an expression is to identify the greatest common factor (GCF) of all the terms. The GCF is the largest expression that divides all the terms evenly.

In this case, the GCF of the given expression is 3t^2.

import sympy as sp

t = sp.symbols('t')



expr = 27t**3 - 36t**2 - 12*t + 16



gcf = sp.gcd(expr, t**2)
print(gcf)

Step 2: Factor the Expression

Now that we have factored out the GCF, we can focus on factoring the remaining expression.

The expression can be rewritten as:

3t2(9t−12)−4(9t−12)3t^2(9t - 12) - 4(9t - 12)

We can see that the expression is a difference of squares, which can be factored as:

(3t2−4)(9t−12)(3t^2 - 4)(9t - 12) 

# Factor the expression
factored_expr = sp.factor(expr)
print(factored_expr)

Step 3: Write the Factored Form

Now that we have factored the expression, we can write it in the desired form:

(3t2−4)(9t−12)(3t^2 - 4)(9t - 12)

This is the factored form of the given expression.

Conclusion

Factoring expressions is a crucial skill in algebra that involves breaking down an expression into simpler components. In this article, we explored the factored form of a given expression and provided a step-by-step guide on how to factor it. We identified the greatest common factor (GCF) of the expression, factored the remaining expression, and wrote the factored form in the desired form.

Final Answer

The factored form of the expression is:

(3t^2 - 4)(9t - 12)$<br/> **Factoring Expressions: A Q&A Guide** =====================================

Introduction

Factoring expressions is a fundamental concept in algebra that involves breaking down an expression into simpler components. In our previous article, we explored the factored form of a given expression and provided a step-by-step guide on how to factor it. In this article, we will answer some frequently asked questions about factoring expressions.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest expression that divides all the terms of an expression evenly. It is the first step in factoring an expression.

Q: How do I identify the GCF of an expression?

A: To identify the GCF of an expression, you need to find the largest expression that divides all the terms of the expression evenly. You can use the following steps:

  1. List all the factors of each term in the expression.
  2. Identify the common factors among all the terms.
  3. Choose the largest common factor as the GCF.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

a2−b2=(a+b)(a−b)</span></p><p>Thisformulacanbeusedtofactorexpressionsthatareintheformofadifferenceofsquares.</p><h2><strong>Q:HowdoIfactoradifferenceofsquares?</strong></h2><p>A:Tofactoradifferenceofsquares,youneedtousethedifferenceofsquaresformula:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mostretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mostretchy="false">)</mo><mostretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">a2−b2=(a+b)(a−b)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.9474em;vertical−align:−0.0833em;"></span><spanclass="mord"><spanclass="mordmathnormal">a</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8641em;"></span><spanclass="mord"><spanclass="mordmathnormal">b</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mordmathnormal">a</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">b</span><spanclass="mclose">)</span><spanclass="mopen">(</span><spanclass="mordmathnormal">a</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">b</span><spanclass="mclose">)</span></span></span></span></span></p><p>Forexample,ifyouhavetheexpression:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow><annotationencoding="application/x−tex">x2−4</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.9474em;vertical−align:−0.0833em;"></span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">4</span></span></span></span></span></p><p>Youcanfactoritas:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mostretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mostretchy="false">)</mo><mostretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(x+2)(x−2)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">2</span><spanclass="mclose">)</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">2</span><spanclass="mclose">)</span></span></span></span></span></p><h2><strong>Q:Whatisthegreatestcommondivisor(GCD)oftwoexpressions?</strong></h2><p>A:Thegreatestcommondivisor(GCD)oftwoexpressionsisthelargestexpressionthatdividesbothexpressionsevenly.ItissimilartotheGCF,butitisusedfortwoexpressionsinsteadofone.</p><h2><strong>Q:HowdoIfindtheGCDoftwoexpressions?</strong></h2><p>A:TofindtheGCDoftwoexpressions,youneedtousethefollowingsteps:</p><ol><li>Listallthefactorsofeachexpression.</li><li>Identifythecommonfactorsamongbothexpressions.</li><li>ChoosethelargestcommonfactorastheGCD.</li></ol><h2><strong>Q:Whatistheleastcommonmultiple(LCM)oftwoexpressions?</strong></h2><p>A:Theleastcommonmultiple(LCM)oftwoexpressionsisthesmallestexpressionthatisamultipleofbothexpressions.ItistheoppositeoftheGCD.</p><h2><strong>Q:HowdoIfindtheLCMoftwoexpressions?</strong></h2><p>A:TofindtheLCMoftwoexpressions,youneedtousethefollowingsteps:</p><ol><li>Listallthemultiplesofeachexpression.</li><li>Identifythesmallestcommonmultipleamongbothexpressions.</li><li>ChoosethesmallestcommonmultipleastheLCM.</li></ol><h2><strong>Conclusion</strong></h2><p>Factoringexpressionsisacrucialskillinalgebrathatinvolvesbreakingdownanexpressionintosimplercomponents.Inthisarticle,weansweredsomefrequentlyaskedquestionsaboutfactoringexpressions,includingthegreatestcommonfactor(GCF),differenceofsquaresformula,greatestcommondivisor(GCD),andleastcommonmultiple(LCM).</p><h2><strong>FinalAnswer</strong></h2><p>Thefactoredformoftheexpressionis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mostretchy="false">(</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mostretchy="false">)</mo><mostretchy="false">(</mo><mn>9</mn><mi>t</mi><mo>−</mo><mn>12</mn><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(3t2−4)(9t−12)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">3</span><spanclass="mord"><spanclass="mordmathnormal">t</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">4</span><spanclass="mclose">)</span><spanclass="mopen">(</span><spanclass="mord">9</span><spanclass="mordmathnormal">t</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">12</span><spanclass="mclose">)</span></span></span></span></span></p><p>Wehopethisarticlehashelpedyouunderstandtheconceptoffactoringexpressionsandhowtofactorthem.Ifyouhaveanymorequestions,feelfreetoask!</p>a^2 - b^2 = (a + b)(a - b) </span></p> <p>This formula can be used to factor expressions that are in the form of a difference of squares.</p> <h2><strong>Q: How do I factor a difference of squares?</strong></h2> <p>A: To factor a difference of squares, you need to use the difference of squares formula:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a^2 - b^2 = (a + b)(a - b) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><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 mathnormal">b</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><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 mathnormal">b</span><span class="mclose">)</span></span></span></span></span></p> <p>For example, if you have the expression:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">x^2 - 4 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span></span></p> <p>You can factor it as:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x + 2)(x - 2) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><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">2</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><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">2</span><span class="mclose">)</span></span></span></span></span></p> <h2><strong>Q: What is the greatest common divisor (GCD) of two expressions?</strong></h2> <p>A: The greatest common divisor (GCD) of two expressions is the largest expression that divides both expressions evenly. It is similar to the GCF, but it is used for two expressions instead of one.</p> <h2><strong>Q: How do I find the GCD of two expressions?</strong></h2> <p>A: To find the GCD of two expressions, you need to use the following steps:</p> <ol> <li>List all the factors of each expression.</li> <li>Identify the common factors among both expressions.</li> <li>Choose the largest common factor as the GCD.</li> </ol> <h2><strong>Q: What is the least common multiple (LCM) of two expressions?</strong></h2> <p>A: The least common multiple (LCM) of two expressions is the smallest expression that is a multiple of both expressions. It is the opposite of the GCD.</p> <h2><strong>Q: How do I find the LCM of two expressions?</strong></h2> <p>A: To find the LCM of two expressions, you need to use the following steps:</p> <ol> <li>List all the multiples of each expression.</li> <li>Identify the smallest common multiple among both expressions.</li> <li>Choose the smallest common multiple as the LCM.</li> </ol> <h2><strong>Conclusion</strong></h2> <p>Factoring expressions is a crucial skill in algebra that involves breaking down an expression into simpler components. In this article, we answered some frequently asked questions about factoring expressions, including the greatest common factor (GCF), difference of squares formula, greatest common divisor (GCD), and least common multiple (LCM).</p> <h2><strong>Final Answer</strong></h2> <p>The factored form of the expression is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>9</mn><mi>t</mi><mo>−</mo><mn>12</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(3t^2 - 4)(9t - 12) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><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">4</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord">9</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><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">12</span><span class="mclose">)</span></span></span></span></span></p> <p>We hope this article has helped you understand the concept of factoring expressions and how to factor them. If you have any more questions, feel free to ask!</p>