Divide. If There Is A Remainder, Include It As A Simplified Fraction. ( − A 3 − 4 A 2 ) ÷ A 2 (-a^3-4a^2) \div A^2 ( − A 3 − 4 A 2 ) ÷ A 2 □ \square □

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Introduction

When it comes to dividing algebraic expressions, it's essential to understand the rules and procedures involved. In this article, we will focus on dividing a polynomial expression by another polynomial expression, specifically the division of $(-a^3-4a^2)$ by $a^2$. We will use the concept of polynomial long division and synthetic division to simplify the expression and find the quotient and remainder.

Understanding Polynomial Long Division

Polynomial long division is a method used to divide a polynomial expression by another polynomial expression. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.

Dividing $(-a^3-4a^2)$ by $a^2$

To divide $(-a^3-4a^2)$ by $a^2$, we can use polynomial long division. We start by dividing the highest degree term of the dividend, $-a^3$, by the highest degree term of the divisor, $a^2$. This gives us $-a$. We then multiply the entire divisor, $a^2$, by $-a$ and subtract it from the dividend.

Step 1: Divide the highest degree term of the dividend by the highest degree term of the divisor

$$\frac{-a^3}{a^2} = -a$$

Step 2: Multiply the entire divisor by the result and subtract it from the dividend

$$(-a^3-4a^2) - (-a^3-2a^2) = -2a^2$$

Step 3: Repeat the process until the degree of the remainder is less than the degree of the divisor

We now have a new dividend, $-2a^2$. We can divide the highest degree term of the new dividend, $-2a^2$, by the highest degree term of the divisor, $a^2$. This gives us $-2$. We then multiply the entire divisor, $a^2$, by $-2$ and subtract it from the new dividend.

Step 4: Divide the highest degree term of the new dividend by the highest degree term of the divisor

$$\frac{-2a^2}{a^2} = -2$$

Step 5: Multiply the entire divisor by the result and subtract it from the new dividend

$$(-2a^2) - (-2a^2) = 0$$

Conclusion

We have successfully divided $(-a^3-4a^2)$ by $a^2$ using polynomial long division. The quotient is $-a-2$ and the remainder is $0$. This means that $(-a^3-4a^2)$ can be expressed as $(-a-2)(a^2)$.

Simplifying the Expression

We can simplify the expression $(-a-2)(a^2)$ by multiplying the two binomials together.

Step 1: Multiply the two binomials together

$$(a^2)(-a) + (a^2)(-2) = -a^3 - 2a^2$$

Step 2: Simplify the expression

$$-a^3 - 2a^2 = -a^3 - 4a^2$$

Conclusion

We have successfully simplified the expression $(-a-2)(a^2)$ by multiplying the two binomials together. The simplified expression is $-a^3 - 4a^2$.

Final Answer

The final answer is $\boxed{-a-2}$.

Discussion

In this article, we have discussed the division of a polynomial expression by another polynomial expression. We have used the concept of polynomial long division and synthetic division to simplify the expression and find the quotient and remainder. We have also simplified the expression by multiplying the two binomials together. The final answer is $\boxed{-a-2}$.

Related Topics

• Polynomial long division
• Synthetic division
• Dividing polynomial expressions
• Simplifying polynomial expressions

References

• [1] "Polynomial Long Division" by Math Open Reference
• [2] "Synthetic Division" by Mathway
• [3] "Dividing Polynomial Expressions" by Khan Academy
• [4] "Simplifying Polynomial Expressions" by Purplemath
  

Introduction

In our previous article, we discussed the division of a polynomial expression by another polynomial expression. We used the concept of polynomial long division and synthetic division to simplify the expression and find the quotient and remainder. In this article, we will answer some frequently asked questions related to dividing polynomial expressions.

Q1: What is polynomial long division?

A1: Polynomial long division is a method used to divide a polynomial expression by another polynomial expression. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend.

Q2: How do I divide a polynomial expression by another polynomial expression?

A2: To divide a polynomial expression by another polynomial expression, you can use polynomial long division or synthetic division. Polynomial long division involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. Synthetic division involves using a shortcut method to divide the polynomial expression.

Q3: What is the quotient and remainder in polynomial division?

A3: The quotient is the result of dividing the dividend by the divisor, and the remainder is the amount left over after the division. In polynomial division, the quotient is a polynomial expression and the remainder is a polynomial expression of a lower degree than the divisor.

Q4: How do I simplify a polynomial expression after dividing?

A4: To simplify a polynomial expression after dividing, you can multiply the quotient by the divisor and add the remainder. This will give you the original polynomial expression.

Q5: What are some common mistakes to avoid when dividing polynomial expressions?

A5: Some common mistakes to avoid when dividing polynomial expressions include:

• Not following the order of operations
• Not multiplying the entire divisor by the result
• Not subtracting the result from the dividend
• Not simplifying the expression after dividing

Q6: Can I use synthetic division to divide polynomial expressions?

A6: Yes, you can use synthetic division to divide polynomial expressions. Synthetic division is a shortcut method that involves using a single row of numbers to divide the polynomial expression.

Q7: How do I determine the degree of a polynomial expression?

A7: To determine the degree of a polynomial expression, you can count the number of terms in the expression and the highest power of the variable in each term.

Q8: Can I divide a polynomial expression by a non-polynomial expression?

A8: No, you cannot divide a polynomial expression by a non-polynomial expression. Polynomial division is only used to divide polynomial expressions by other polynomial expressions.

Q9: How do I check my work when dividing polynomial expressions?

A9: To check your work when dividing polynomial expressions, you can multiply the quotient by the divisor and add the remainder. This will give you the original polynomial expression.

Q10: What are some real-world applications of polynomial division?

A10: Polynomial division has many real-world applications, including:

• Calculating the area and perimeter of shapes
• Finding the volume of solids
• Modeling population growth
• Analyzing data

Conclusion

In this article, we have answered some frequently asked questions related to dividing polynomial expressions. We have discussed the concept of polynomial long division and synthetic division, and provided examples of how to divide polynomial expressions. We have also discussed some common mistakes to avoid and how to check your work when dividing polynomial expressions.

Final Answer

The final answer is $\boxed{0}$.

Discussion

In this article, we have discussed the division of polynomial expressions and provided answers to some frequently asked questions. We have also discussed some real-world applications of polynomial division. If you have any further questions or need additional help, please don't hesitate to ask.

Related Topics

• Polynomial long division
• Synthetic division
• Dividing polynomial expressions
• Simplifying polynomial expressions

References

• [1] "Polynomial Long Division" by Math Open Reference
• [2] "Synthetic Division" by Mathway
• [3] "Dividing Polynomial Expressions" by Khan Academy
• [4] "Simplifying Polynomial Expressions" by Purplemath