Simplify The Expression:${ \left(4 A^4 B^5 C + 5 A^3 B^5 C^2 + 6 B^2\right) \times \left(-5 A^4 B + A C^3 - 5 A B C^2\right) }$

Introduction

In algebra, simplifying expressions is a crucial skill that helps in solving complex equations and problems. It involves manipulating the given expression to make it more manageable and easier to work with. In this article, we will focus on simplifying a given expression using algebraic techniques. We will break down the expression into smaller parts, apply various algebraic rules, and then combine the results to obtain the simplified form.

Understanding the Expression

The given expression is a product of two polynomials:

$$\left(4 a^4 b^5 c + 5 a^3 b^5 c^2 + 6 b^2\right) \times \left(-5 a^4 b + a c^3 - 5 a b c^2\right)$$

To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, we have:

$$a(b + c) = ab + ac$$

Distributing the First Polynomial

We will start by distributing the first polynomial to each term of the second polynomial. This will result in a sum of products, which we can then simplify further.

$$\left(4 a^4 b^5 c + 5 a^3 b^5 c^2 + 6 b^2\right) \times \left(-5 a^4 b + a c^3 - 5 a b c^2\right)$$

$$= 4 a^4 b^5 c \times (-5 a^4 b) + 4 a^4 b^5 c \times (a c^3) + 4 a^4 b^5 c \times (-5 a b c^2)$$

$$+ 5 a^3 b^5 c^2 \times (-5 a^4 b) + 5 a^3 b^5 c^2 \times (a c^3) + 5 a^3 b^5 c^2 \times (-5 a b c^2)$$

$$+ 6 b^2 \times (-5 a^4 b) + 6 b^2 \times (a c^3) + 6 b^2 \times (-5 a b c^2)$$

Simplifying the Products

Now, we will simplify each product by combining like terms and applying the rules of exponents.

$$= -20 a^8 b^6 c - 4 a^5 b^5 c^4 - 20 a^5 b^6 c^3$$

$$- 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

Combining Like Terms

Next, we will combine like terms by adding or subtracting the coefficients of the same variables.

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

Final Simplification

After combining like terms, we can simplify the expression further by factoring out common terms.

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

$$- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^4$$

$$= -20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3$$

- <br/> # Simplify the Expression: A Comprehensive Guide to Algebraic Manipulation

Q&A: Simplifying the Expression

Q: What is the given expression?

A: The given expression is a product of two polynomials:

$$\left(4 a^4 b^5 c + 5 a^3 b^5 c^2 + 6 b^2\right) \times \left(-5 a^4 b + a c^3 - 5 a b c^2\right) </span></p> <h3>Q: What is the first step in simplifying the expression?</h3> <p>A: The first step in simplifying the expression is to distribute the first polynomial to each term of the second polynomial.</p> <h3>Q: How do we distribute the first polynomial?</h3> <p>A: We distribute the first polynomial by multiplying each term of the first polynomial to each term of the second polynomial.</p> <h3>Q: What is the result of distributing the first polynomial?</h3> <p>A: The result of distributing the first polynomial is a sum of products, which we can then simplify further.</p> <h3>Q: How do we simplify the products?</h3> <p>A: We simplify the products by combining like terms and applying the rules of exponents.</p> <h3>Q: What is the final simplified expression?</h3> <p>A: The final simplified 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>−</mo><mn>20</mn><msup><mi>a</mi><mn>8</mn></msup><msup><mi>b</mi><mn>6</mn></msup><mi>c</mi><mo>−</mo><mn>20</mn><msup><mi>a</mi><mn>5</mn></msup><msup><mi>b</mi><mn>6</mn></msup><msup><mi>c</mi><mn>3</mn></msup><mo>−</mo><mn>25</mn><msup><mi>a</mi><mn>7</mn></msup><msup><mi>b</mi><mn>6</mn></msup><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><msup><mi>a</mi><mn>4</mn></msup><msup><mi>b</mi><mn>5</mn></msup><msup><mi>c</mi><mn>4</mn></msup><mo>+</mo><mn>25</mn><msup><mi>a</mi><mn>4</mn></msup><msup><mi>b</mi><mn>6</mn></msup><msup><mi>c</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">-20 a^8 b^6 c - 20 a^5 b^6 c^3 - 25 a^7 b^6 c^2 + 5 a^4 b^5 c^4 + 25 a^4 b^6 c^3 </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><span class="mord">20</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">8</span></span></span></span></span></span></span></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">6</span></span></span></span></span></span></span></span><span class="mord mathnormal">c</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:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">20</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">5</span></span></span></span></span></span></span></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">6</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">c</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">3</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:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">25</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">7</span></span></span></span></span></span></span></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">6</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">c</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:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">5</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">4</span></span></span></span></span></span></span></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">5</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">c</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">4</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">25</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">4</span></span></span></span></span></span></span></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">6</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">c</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">3</span></span></span></span></span></span></span></span></span></span></span></span></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>−</mo><mn>30</mn><msup><mi>a</mi><mn>5</mn></msup><msup><mi>b</mi><mn>6</mn></msup><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><msup><mi>a</mi><mn>3</mn></msup><msup><mi>b</mi><mn>2</mn></msup><msup><mi>c</mi><mn>3</mn></msup><mo>−</mo><mn>30</mn><msup><mi>a</mi><mn>4</mn></msup><msup><mi>b</mi><mn>3</mn></msup><msup><mi>c</mi><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">- 30 a^5 b^6 c^2 + 6 a^3 b^2 c^3 - 30 a^4 b^3 c^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><span class="mord">30</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">5</span></span></span></span></span></span></span></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">6</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">c</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:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">6</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">3</span></span></span></span></span></span></span></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="mord"><span class="mord mathnormal">c</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">3</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">30</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">4</span></span></span></span></span></span></span></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">3</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">c</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">4</span></span></span></span></span></span></span></span></span></span></span></span></p> <h3>Q: What is the purpose of simplifying the expression?</h3> <p>A: The purpose of simplifying the expression is to make it more manageable and easier to work with.</p> <h3>Q: How does simplifying the expression help in solving complex equations and problems?</h3> <p>A: Simplifying the expression helps in solving complex equations and problems by making it easier to identify patterns and relationships between variables.</p> <h3>Q: What are some common techniques used in simplifying expressions?</h3> <p>A: Some common techniques used in simplifying expressions include:</p> <ul> <li>Distributing polynomials</li> <li>Combining like terms</li> <li>Applying the rules of exponents</li> <li>Factoring out common terms</li> </ul> <h3>Q: How can we apply these techniques to simplify more complex expressions?</h3> <p>A: We can apply these techniques to simplify more complex expressions by breaking them down into smaller parts, identifying patterns and relationships between variables, and using algebraic rules to simplify the expression.</p> <h3>Q: What are some common mistakes to avoid when simplifying expressions?</h3> <p>A: Some common mistakes to avoid when simplifying expressions include:</p> <ul> <li>Failing to distribute polynomials correctly</li> <li>Failing to combine like terms</li> <li>Failing to apply the rules of exponents correctly</li> <li>Failing to factor out common terms</li> </ul> <h3>Q: How can we avoid these mistakes?</h3> <p>A: We can avoid these mistakes by carefully reading and following the instructions, using algebraic rules and techniques correctly, and double-checking our work.</p> <h2>Conclusion</h2> <p>Simplifying expressions is a crucial skill in algebra that helps in solving complex equations and problems. By understanding the techniques and rules involved in simplifying expressions, we can make it easier to identify patterns and relationships between variables, and solve complex problems with confidence.</p>$$