Select All That Are Polynomials.A. $49x^2 - 64$B. $-7xy + \frac{2}{y}$C. $45y^2 + Y - Y^3 + 8 + X^3$D. -16E. $3x^{-2} + 6x^2$

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Introduction

In mathematics, polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. They are a fundamental concept in algebra and are used to model various real-world phenomena. In this article, we will explore the concept of polynomials and learn how to identify them.

What are Polynomials?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are typically represented by letters such as x, y, or z, and the coefficients are numbers that are multiplied with the variables. Polynomials can have one or more terms, and each term must be a product of a coefficient and a variable or a constant.

Examples of Polynomials

Let's consider some examples of polynomials:

  • 2x + 3: This is a polynomial with one term, where 2 is the coefficient and x is the variable.
  • x^2 + 4x - 5: This is a polynomial with three terms, where x^2 is the first term, 4x is the second term, and -5 is the third term.
  • 3y^2 - 2y + 1: This is a polynomial with three terms, where 3y^2 is the first term, -2y is the second term, and 1 is the third term.

Selecting Polynomials from the Given Options

Now, let's apply our knowledge of polynomials to select the polynomials from the given options.

A. 49x2−6449x^2 - 64

This expression consists of two terms: 49x249x^2 and −64-64. Both terms are products of coefficients and variables, and there are no addition or subtraction operations between them. Therefore, this expression is a polynomial.

B. −7xy+2y-7xy + \frac{2}{y}

This expression consists of two terms: −7xy-7xy and 2y\frac{2}{y}. However, the second term contains a fraction, which is not a polynomial. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication. Therefore, this expression is not a polynomial.

C. 45y2+y−y3+8+x345y^2 + y - y^3 + 8 + x^3

This expression consists of four terms: 45y245y^2, yy, −y3-y^3, and 88, as well as x3x^3. However, the expression contains addition and subtraction operations between terms, which is allowed in polynomials. Therefore, this expression is a polynomial.

D. -16

This expression consists of a single term, which is a constant. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication. Therefore, this expression is not a polynomial.

E. 3x−2+6x23x^{-2} + 6x^2

This expression consists of two terms: 3x−23x^{-2} and 6x26x^2. However, the first term contains a negative exponent, which is not allowed in polynomials. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication. Therefore, this expression is not a polynomial.

Conclusion

In conclusion, the polynomials from the given options are:

  • A. 49x2−6449x^2 - 64
  • C. 45y2+y−y3+8+x345y^2 + y - y^3 + 8 + x^3

These expressions consist of variables and coefficients combined using only addition, subtraction, and multiplication, which are the defining characteristics of polynomials.

Frequently Asked Questions

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. A rational expression, on the other hand, is an expression consisting of a fraction of two polynomials.

Q: Can a polynomial have a negative exponent?

A: No, a polynomial cannot have a negative exponent. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: Can a polynomial have a fraction as a coefficient?

A: No, a polynomial cannot have a fraction as a coefficient. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication.

References

Glossary

  • Coefficient: A number that is multiplied with a variable in a polynomial.
  • Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
  • Term: A product of a coefficient and a variable or a constant in a polynomial.
  • Variable: A letter or symbol that represents a value in a polynomial.
    Polynomial Q&A: Frequently Asked Questions and Answers ===========================================================

Introduction

Polynomials are a fundamental concept in mathematics, and they have numerous applications in various fields such as algebra, calculus, and engineering. However, many students and professionals often struggle to understand the concept of polynomials and how to identify them. In this article, we will provide answers to frequently asked questions about polynomials, helping you to better understand this important mathematical concept.

Q&A

Q: What is a polynomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: What are the characteristics of a polynomial?

A: A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication. It can have one or more terms, and each term must be a product of a coefficient and a variable or a constant.

Q: Can a polynomial have a variable with a negative exponent?

A: No, a polynomial cannot have a variable with a negative exponent. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: Can a polynomial have a fraction as a coefficient?

A: No, a polynomial cannot have a fraction as a coefficient. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: Can a polynomial have a variable with a fractional exponent?

A: No, a polynomial cannot have a variable with a fractional exponent. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: Can a polynomial have a constant term?

A: Yes, a polynomial can have a constant term. A constant term is a term that does not contain any variables.

Q: Can a polynomial have multiple variables?

A: Yes, a polynomial can have multiple variables. For example, the polynomial x2+2xy+y2x^2 + 2xy + y^2 has two variables, x and y.

Q: Can a polynomial have a variable raised to a power?

A: Yes, a polynomial can have a variable raised to a power. For example, the polynomial x2+3x+2x^2 + 3x + 2 has a variable raised to the power of 2.

Q: Can a polynomial have a constant raised to a power?

A: Yes, a polynomial can have a constant raised to a power. For example, the polynomial 22+3x+42^2 + 3x + 4 has a constant raised to the power of 2.

Q: Can a polynomial have a term with a negative coefficient?

A: Yes, a polynomial can have a term with a negative coefficient. For example, the polynomial −x2+3x+2-x^2 + 3x + 2 has a term with a negative coefficient.

Q: Can a polynomial have a term with a zero coefficient?

A: Yes, a polynomial can have a term with a zero coefficient. For example, the polynomial x2+0x+2x^2 + 0x + 2 has a term with a zero coefficient.

Q: Can a polynomial have a term with a variable raised to a power of zero?

A: Yes, a polynomial can have a term with a variable raised to a power of zero. For example, the polynomial x2+3x+1x^2 + 3x + 1 has a term with a variable raised to a power of zero.

Q: Can a polynomial have a term with a constant raised to a power of zero?

A: Yes, a polynomial can have a term with a constant raised to a power of zero. For example, the polynomial 22+3x+12^2 + 3x + 1 has a term with a constant raised to a power of zero.

Q: Can a polynomial have a term with a variable and a constant?

A: Yes, a polynomial can have a term with a variable and a constant. For example, the polynomial x2+3x+2x^2 + 3x + 2 has a term with a variable and a constant.

Q: Can a polynomial have a term with a constant and a variable?

A: Yes, a polynomial can have a term with a constant and a variable. For example, the polynomial 2x2+3x+12x^2 + 3x + 1 has a term with a constant and a variable.

Q: Can a polynomial have a term with a variable and a constant raised to a power?

A: Yes, a polynomial can have a term with a variable and a constant raised to a power. For example, the polynomial x2+3x2+2x^2 + 3x^2 + 2 has a term with a variable and a constant raised to a power.

Q: Can a polynomial have a term with a constant and a variable raised to a power?

A: Yes, a polynomial can have a term with a constant and a variable raised to a power. For example, the polynomial 2x2+3x2+12x^2 + 3x^2 + 1 has a term with a constant and a variable raised to a power.

Q: Can a polynomial have a term with a variable and a constant raised to different powers?

A: Yes, a polynomial can have a term with a variable and a constant raised to different powers. For example, the polynomial x2+3x+2x^2 + 3x + 2 has a term with a variable raised to the power of 2 and a constant raised to the power of 1.

Q: Can a polynomial have a term with a constant and a variable raised to different powers?

A: Yes, a polynomial can have a term with a constant and a variable raised to different powers. For example, the polynomial 2x2+3x+12x^2 + 3x + 1 has a term with a constant raised to the power of 1 and a variable raised to the power of 2.

Q: Can a polynomial have a term with a variable and a constant raised to the same power?

A: Yes, a polynomial can have a term with a variable and a constant raised to the same power. For example, the polynomial x2+3x2+2x^2 + 3x^2 + 2 has a term with a variable and a constant raised to the power of 2.

Q: Can a polynomial have a term with a constant and a variable raised to the same power?

A: Yes, a polynomial can have a term with a constant and a variable raised to the same power. For example, the polynomial 2x2+3x2+12x^2 + 3x^2 + 1 has a term with a constant and a variable raised to the power of 2.

Conclusion

In conclusion, polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. They can have one or more terms, and each term must be a product of a coefficient and a variable or a constant. Polynomials can have variables with positive or negative exponents, fractions as coefficients, and constants raised to powers. They can also have terms with variables and constants raised to different powers or the same power.

Frequently Asked Questions

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. A rational expression, on the other hand, is an expression consisting of a fraction of two polynomials.

Q: Can a polynomial have a negative exponent?

A: No, a polynomial cannot have a negative exponent. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: Can a polynomial have a fraction as a coefficient?

A: No, a polynomial cannot have a fraction as a coefficient. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: Can a polynomial have a variable with a fractional exponent?

A: No, a polynomial cannot have a variable with a fractional exponent. A polynomial must consist of variables and coefficients combined using only addition, subtraction, and multiplication.

References

Glossary

  • Coefficient: A number that is multiplied with a variable in a polynomial.
  • Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
  • Term: A product of a coefficient and a variable or a constant in a polynomial.
  • Variable: A letter or symbol that represents a value in a polynomial.