What Is The Sum Of The Polynomials?$\[ \begin{array}{r} 17m - 12n - 1 \\ + \quad 4 - 13m - 12n \\ \hline \end{array} \\]A. \[$4m + 3\$\]B. \[$4m - 24n + 3\$\]C. \[$30m - 5\$\]D. \[$30m - 24n - 5\$\]

Understanding Polynomials and Their Operations

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. In this article, we will explore the concept of summing polynomials, which involves combining two or more polynomials using addition and subtraction operations.

The Basics of Polynomial Addition and Subtraction

When adding or subtracting polynomials, we combine like terms, which are terms that have the same variable(s) raised to the same power. For example, in the polynomial $3x^2 + 2x - 1$, the terms $3x^2$ and $2x$ are like terms because they both have the variable $x$ raised to the power of 2.

The Problem: Summing Two Polynomials

Let's consider the problem of summing the two polynomials:

$$\begin{array}{r} 17m - 12n - 1 \\ + \quad 4 - 13m - 12n \\ \hline \end{array}$$

To find the sum of these polynomials, we need to combine like terms. We can start by adding the coefficients of the like terms.

Step 1: Combine Like Terms

The first polynomial has the term $17m$, and the second polynomial has the term $-13m$. We can combine these two terms by adding their coefficients:

$$17m + (-13m) = 4m$$

The first polynomial also has the term $-12n$, and the second polynomial has the term $-12n$. We can combine these two terms by adding their coefficients:

$$-12n + (-12n) = -24n$$

The first polynomial has a constant term of $-1$, and the second polynomial has a constant term of $4$. We can combine these two terms by adding them:

$$-1 + 4 = 3$$

Step 2: Write the Sum of the Polynomials

Now that we have combined the like terms, we can write the sum of the polynomials:

$$4m - 24n + 3$$

Conclusion

In this article, we explored the concept of summing polynomials and applied it to a specific problem. We learned how to combine like terms and write the sum of two polynomials. The correct answer to the problem is:

$$4m - 24n + 3$$

This is option B in the given choices.

Key Takeaways

• Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• When adding or subtracting polynomials, we combine like terms, which are terms that have the same variable(s) raised to the same power.
• To find the sum of two polynomials, we need to combine like terms by adding their coefficients.
• The sum of two polynomials can be written by combining the like terms and writing the resulting expression.

Frequently Asked Questions

• What is the sum of the polynomials $17m - 12n - 1$ and $4 - 13m - 12n$?
  • The sum of the polynomials is $4m - 24n + 3$.
• How do we combine like terms in polynomials?
  • We combine like terms by adding their coefficients.
• What is the result of combining the terms $17m$ and $-13m$?
  • The result is $4m$.
• What is the result of combining the terms $-12n$ and $-12n$?
  • The result is $-24n$.

References

• Polynomial Addition and Subtraction
• Like Terms
• Polynomial Operations
  Polynomial Summation Q&A ==========================

Frequently Asked Questions

Q: What is the sum of the polynomials $17m - 12n - 1$ and $4 - 13m - 12n$?

A: The sum of the polynomials is $4m - 24n + 3$.

Q: How do we combine like terms in polynomials?

A: We combine like terms by adding their coefficients.

Q: What is the result of combining the terms $17m$ and $-13m$?

A: The result is $4m$.

Q: What is the result of combining the terms $-12n$ and $-12n$?

A: The result is $-24n$.

Q: Can we add polynomials with different variables?

A: No, we can only add polynomials with the same variables raised to the same power.

Q: How do we subtract polynomials?

A: To subtract a polynomial, we change the sign of each term in the polynomial and then add the resulting polynomials.

Q: What is the sum of the polynomials $2x^2 + 3x - 1$ and $x^2 - 2x + 3$?

A: To find the sum, we combine like terms:

• $2x^2 + x^2 = 3x^2$
• $3x - 2x = x$
• $-1 + 3 = 2$

The sum of the polynomials is $3x^2 + x + 2$.

Q: What is the difference between the polynomials $2x^2 + 3x - 1$ and $x^2 - 2x + 3$?

A: To find the difference, we change the sign of each term in the second polynomial and then add the resulting polynomials:

• $2x^2 + x^2 = 3x^2$
• $3x + 2x = 5x$
• $-1 - 3 = -4$

The difference between the polynomials is $3x^2 + 5x - 4$.

Q: Can we multiply polynomials?

A: Yes, we can multiply polynomials using the distributive property.

Q: What is the product of the polynomials $2x^2 + 3x - 1$ and $x^2 - 2x + 3$?

A: To find the product, we multiply each term in the first polynomial by each term in the second polynomial and then combine like terms:

• $(2x^2)(x^2) = 2x^4$
• $(2x^2)(-2x) = -4x^3$
• $(2x^2)(3) = 6x^2$
• $(3x)(x^2) = 3x^3$
• $(3x)(-2x) = -6x^2$
• $(3x)(3) = 9x$
• $(-1)(x^2) = -x^2$
• $(-1)(-2x) = 2x$
• $(-1)(3) = -3$

Combining like terms, we get:

• $2x^4 - 4x^3 + 6x^2 + 3x^3 - 6x^2 + 9x - x^2 + 2x - 3$
• $2x^4 - x^3 + 9x - 3$

The product of the polynomials is $2x^4 - x^3 + 9x - 3$.

Q: Can we divide polynomials?

A: Yes, we can divide polynomials using long division or synthetic division.

Q: What is the quotient of the polynomials $2x^3 + 3x^2 - 1$ and $x + 2$?

A: To find the quotient, we use long division:

• Divide $2x^3$ by $x$ to get $2x^2$
• Multiply $x + 2$ by $2x^2$ to get $2x^3 + 4x^2$
• Subtract $2x^3 + 4x^2$ from $2x^3 + 3x^2 - 1$ to get $-x^2 - 1$
• Divide $-x^2$ by $x$ to get $-x$
• Multiply $x + 2$ by $-x$ to get $-x^2 - 2x$
• Subtract $-x^2 - 2x$ from $-x^2 - 1$ to get $2x - 1$
• Divide $2x$ by $x$ to get $2$
• Multiply $x + 2$ by $2$ to get $2x + 4$
• Subtract $2x + 4$ from $2x - 1$ to get $-5$

The quotient of the polynomials is $2x^2 - x + 2$ with a remainder of $-5$.

Q: What is the remainder of the polynomials $2x^3 + 3x^2 - 1$ and $x + 2$?

A: The remainder is $-5$.

Q: Can we simplify polynomials?

A: Yes, we can simplify polynomials by combining like terms and factoring out common factors.

Q: What is the simplified form of the polynomial $2x^3 + 3x^2 - 1$?

A: We can simplify the polynomial by combining like terms:

• $2x^3 + 3x^2 - 1 = 2x^3 + 3x^2 - 1$

However, we can factor out a common factor of $x^2$:

• $2x^3 + 3x^2 - 1 = x^2(2x + 3) - 1$

The simplified form of the polynomial is $x^2(2x + 3) - 1$.