The Sum Of Two Polynomials Is − Y Z 2 − 3 Z 2 − 4 Y + 4 -y Z^2-3 Z^2-4 Y+4 − Y Z 2 − 3 Z 2 − 4 Y + 4 . If One Of The Polynomials Is Y − 4 Y Z 2 − 3 Y-4 Y Z^2-3 Y − 4 Y Z 2 − 3 , What Is The Other Polynomial?A. − 2 Y Z 2 − 4 Y + 7 -2 Y Z^2-4 Y+7 − 2 Y Z 2 − 4 Y + 7 B. − 2 Y Z 2 − 3 Y + 1 -2 Y Z^2-3 Y+1 − 2 Y Z 2 − 3 Y + 1 C. − 5 Y Z 2 + 3 Z 2 − 3 Y + 1 -5 Y Z^2+3 Z^2-3 Y+1 − 5 Y Z 2 + 3 Z 2 − 3 Y + 1 D. $3 Y Z^2-3

Introduction

In algebra, polynomials are a fundamental concept used to represent mathematical expressions. When dealing with polynomials, it's essential to understand how to add and subtract them. In this article, we will explore the concept of adding two polynomials and use this knowledge to solve a specific problem.

What are Polynomials?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be written in various forms, including:

• Monomials: A single term with a variable and a coefficient, such as 3x or 2y.
• Binomials: An expression with two terms, such as x + 3 or 2y - 4.
• Polynomials: An expression with three or more terms, such as x + 2y - 3z or 2x^2 + 3y - 4z.

Adding Polynomials

When adding polynomials, we combine like terms, which are terms with the same variable and exponent. For example, when adding 2x^2 + 3y and 4x^2 - 2y, we combine the like terms:

• 2x^2 + 4x^2 = 6x^2
• 3y - 2y = y

The resulting polynomial is 6x^2 + y.

The Problem

We are given the sum of two polynomials: $-y z^2-3 z^2-4 y+4$. One of the polynomials is $y-4 y z^2-3$. We need to find the other polynomial.

Step 1: Identify the Given Polynomial

The given polynomial is $y-4 y z^2-3$. We will use this polynomial to find the other polynomial.

Step 2: Subtract the Given Polynomial from the Sum

To find the other polynomial, we need to subtract the given polynomial from the sum. We will use the distributive property to expand the subtraction:

$-y z^2-3 z^2-4 y+4 - (y-4 y z^2-3)$

Using the distributive property, we get:

$-y z^2-3 z^2-4 y+4 - y + 4 y z^2 + 3$

Combining like terms, we get:

$-y z^2 + 4 y z^2 - 3 z^2 - 3 - 4 y + y + 4$

Simplifying the expression, we get:

$3 y z^2 - 3 z^2 - 3 y + 1$

Conclusion

The other polynomial is $3 y z^2 - 3 z^2 - 3 y + 1$. This polynomial is option C.

Answer

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Q&A: The Sum of Two Polynomials

Q: What is the difference between a polynomial and a monomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A monomial, on the other hand, is a single term with a variable and a coefficient, such as 3x or 2y.

Q: How do you add polynomials?

A: When adding polynomials, we combine like terms, which are terms with the same variable and exponent. For example, when adding 2x^2 + 3y and 4x^2 - 2y, we combine the like terms:

• 2x^2 + 4x^2 = 6x^2
• 3y - 2y = y

The resulting polynomial is 6x^2 + y.

Q: What is the distributive property in algebra?

A: The distributive property is a rule in algebra that allows us to expand the multiplication of a single term over multiple terms. For example, if we have the expression 2(x + 3), we can use the distributive property to expand it as 2x + 6.

Q: How do you subtract polynomials?

A: When subtracting polynomials, we use the distributive property to expand the subtraction. For example, if we have the expression (2x^2 + 3y) - (4x^2 - 2y), we can use the distributive property to expand it as 2x^2 - 4x^2 + 3y + 2y.

Q: What is the difference between a polynomial and a binomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A binomial, on the other hand, is an expression with two terms, such as x + 3 or 2y - 4.

Q: How do you simplify a polynomial expression?

A: To simplify a polynomial expression, we combine like terms and eliminate any unnecessary terms. For example, if we have the expression 2x^2 + 3y - 2x^2 + 2y, we can simplify it by combining the like terms:

• 2x^2 - 2x^2 = 0
• 3y + 2y = 5y

The resulting polynomial is 5y.

Q: What is the sum of the two polynomials $-y z^2-3 z^2-4 y+4$ and $y-4 y z^2-3$?

A: To find the sum of the two polynomials, we add them together:

$-y z^2-3 z^2-4 y+4 + y-4 y z^2-3$

Using the distributive property, we get:

$-y z^2 - 4 y z^2 - 3 z^2 - 3 - 4 y + y + 4$

Simplifying the expression, we get:

$-5 y z^2 - 3 z^2 - 3 y + 1$

Conclusion

In this article, we explored the concept of adding two polynomials and used this knowledge to solve a specific problem. We also answered some common questions related to polynomials and algebra.