Factoring out a greatest common factor

Oct 13, 2025 by windailys.com 0 views

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. Polynomials are a fundamental concept in algebra and are used to model various real-world phenomena. In this article, we will explore which of the given algebraic expressions is a polynomial.

What is a Polynomial?

A polynomial is an expression that can be written in the form:

a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are constants, and $x$ is a variable. The exponents of the variable $x$ are non-negative integers.

Properties of Polynomials

Polynomials have several important properties that distinguish them from other types of expressions. Some of the key properties of polynomials include:

Addition and Subtraction: Polynomials can be added and subtracted using the usual rules of arithmetic.
Multiplication: Polynomials can be multiplied using the distributive property.
Exponents: The exponents of the variable $x$ are non-negative integers.
Coefficients: The coefficients of the polynomial are constants.

Examples of Polynomials

Some examples of polynomials include:

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

Which Algebraic Expression is a Polynomial?

Now that we have a good understanding of what a polynomial is and its properties, let's examine the given algebraic expressions and determine which one is a polynomial.

A. $3m^2n - \frac{2m}{n} + \frac{1}{n}$

This expression is not a polynomial because it contains a fraction with a variable in the denominator. The expression $\frac{2m}{n}$ is not a polynomial because the exponent of the variable $m$ is not a non-negative integer.

B. $\frac{2mn}{5} - \frac{\sqrt{m}}{4} + 4m^5$

This expression is not a polynomial because it contains a square root of a variable. The expression $\sqrt{m}$ is not a polynomial because the exponent of the variable $m$ is not a non-negative integer.

C. $\frac{4m^3}{n^2} - 3mn^5 + \sqrt{8}$

This expression is not a polynomial because it contains a fraction with a variable in the denominator and a square root of a constant. The expression $\frac{4m^3}{n^2}$ is not a polynomial because the exponent of the variable $n$ is not a non-negative integer.

D. $7mn + \frac{3m}{2} + \frac{5n}{4}$

This expression is not a polynomial because it contains fractions with variables in the denominators. The expression $\frac{3m}{2}$ is not a polynomial because the exponent of the variable $m$ is not a non-negative integer.

However, if we simplify the expression by combining the fractions, we get:

\frac{7mn \cdot 4}{4} + \frac{3m}{2} + \frac{5n}{4} = 14mn + \frac{3m}{2} + \frac{5n}{4}

This expression is still not a polynomial because it contains fractions with variables in the denominators.

But, if we multiply the entire expression by the least common multiple of the denominators, which is 4, we get:

4 \cdot (14mn + \frac{3m}{2} + \frac{5n}{4}) = 56mn + 6m + 5n

This expression is a polynomial because it only contains variables and coefficients combined using addition, subtraction, and multiplication, and non-negative integer exponents.

Conclusion

In conclusion, the algebraic expression that is a polynomial is:

56mn + 6m + 5n

This expression meets the definition of a polynomial because it only contains variables and coefficients combined using addition, subtraction, and multiplication, and non-negative integer exponents.

References

Q: What is a polynomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents.

Q: What are the properties of polynomials?

A: Polynomials have several important properties that distinguish them from other types of expressions. Some of the key properties of polynomials include:

Addition and Subtraction: Polynomials can be added and subtracted using the usual rules of arithmetic.
Multiplication: Polynomials can be multiplied using the distributive property.
Exponents: The exponents of the variable $x$ are non-negative integers.
Coefficients: The coefficients of the polynomial are constants.

Q: What are some examples of polynomials?

A: Some examples of polynomials include:

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

Q: How do I determine if an expression is a polynomial?

A: To determine if an expression is a polynomial, you can follow these steps:

Check if the expression contains variables and coefficients combined using only addition, subtraction, and multiplication.
Check if the exponents of the variables are non-negative integers.
Check if the expression contains any fractions or square roots.

If the expression meets these criteria, it is a polynomial.

Q: What is the difference between a polynomial and a rational expression?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, and non-negative integer exponents. A rational expression is an expression that can be written in the form $\frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials.

Q: Can I add or subtract polynomials?

A: Yes, you can add or subtract polynomials using the usual rules of arithmetic.

Q: Can I multiply polynomials?

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

Q: Can I divide polynomials?

A: Yes, you can divide polynomials using polynomial long division.

Q: What is polynomial long division?

A: Polynomial long division is a method for dividing one polynomial by another. It involves dividing the leading term of the dividend by the leading term of the divisor, and then multiplying the result by the divisor and subtracting the product from the dividend.

Q: What is polynomial factorization?

A: Polynomial factorization is the process of expressing a polynomial as a product of simpler polynomials.

Q: Can I factor a polynomial?

A: Yes, you can factor a polynomial using various methods, including:

Factoring out a greatest common factor
Factoring by grouping
Factoring using the difference of squares
Factoring using the sum and difference of cubes

Q: What is the difference of squares?

A: The difference of squares is a method for factoring a polynomial of the form $a^2 - b^2$ , where $a$ and $b$ are constants.

Q: What is the sum and difference of cubes?

A: The sum and difference of cubes is a method for factoring a polynomial of the form $a^3 + b^3$ or $a^3 - b^3$ , where $a$ and $b$ are constants.

Q: Can I use a calculator to factor a polynomial?

A: Yes, you can use a calculator to factor a polynomial. However, it's always a good idea to check the result by hand to make sure it's correct.

Q: What are some common mistakes to avoid when working with polynomials?

A: Some common mistakes to avoid when working with polynomials include:

Forgetting to simplify the expression
Not checking the exponents of the variables
Not checking the coefficients of the polynomial
Not using the correct method for factoring the polynomial

By avoiding these common mistakes, you can ensure that your work with polynomials is accurate and reliable.

Conclusion

In conclusion, polynomials are an important concept in algebra and are used to model various real-world phenomena. By understanding the properties and methods for working with polynomials, you can solve a wide range of problems and make informed decisions.