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POLYNOMIAL

Polynomial

Definition Of Polynomial

A polynomial is a monomial or a sum or difference of two or more monomials.

More About Polynomial

Polynomial function: A polynomial function is a function whose terms each contain a constant multiplied by a power of a variable.
P(x) = a0xn + a1xn-1 + .................+ an-2x2 + an-1x + an is a polynomial function 
where a0, a1, a2,..., an are all real numbers and n is a non-negative integer.
Polynomial in one variable: a0xn + a1xn-1 + .................+ an-2x2 + an-1x + an is a polynomial in one variable.

Video Examples: Adding and Subtracting Polynomials

Example of Polynomial

  • The following are few examples of polynomials.
    2x - 7, 5x2 + 2x - 8, - 7y2 + 4y - 13, 4x2 + 16xy + 16y2

Solved Example on Polynomial

Ques: Simplify the polynomial expression: (10x3 + 12x + 15) - (3x3 + 6x + 10)

Choices:

A. 7x3 + 18x + 25
B. 7x3 + 6x + 5
C. 7x6 + 18x2 + 25
D. 7x6 + 6x2 + 5
Correct Answer: B

Solution:

Step 1: The given polynomial expression (10x3 + 12x + 15) - (3x3 + 6x + 10) can be simplified as 7x3 + 6x + 5. [Combine like terms.]

Quick Summary

  • Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication.
  • The exponents of variables in a polynomial must be non-negative integers.
  • Polynomials can be classified by the number of terms (monomial, binomial, trinomial) and by their degree (highest exponent).
\[ P(x) = a_{0}x^{n} + a_{1}x^{n-1} + \dots + a_{n-2}x^{2} + a_{n-1}x + a_{n} \]

⚠️ Common Mistakes

  • Incorrectly combining terms with different exponents.
  • Forgetting to distribute negative signs when subtracting polynomials.
  • Making errors in arithmetic operations when simplifying expressions.

🌍 Real-World Uses

  • Modeling projectile motion, such as the trajectory of a ball thrown in the air.
  • Designing roller coasters where polynomial functions approximate the curves and slopes for a thrilling and safe ride.
  • Creating models for population growth or decline over time, though often in conjunction with other functions.

📋 Standards Alignment

  • CCSS.MATH.HSA.APR.A.1
  • CCSS.MATH.HSA.APR.B.3

🔗 Related Links

🍎 Teacher Insights

Use visual aids and manipulatives to help students understand the concept of combining like terms. Emphasize the importance of careful organization and attention to detail when simplifying polynomial expressions.

🎓 Prerequisites

  • Variables
  • Exponents
  • Algebraic Expressions
  • Real Numbers

Check Your Knowledge

Q1: Simplify: (2x^2 + 3x - 1) + (x^2 - x + 4)

Q2: Simplify: (10x^3 + 12x + 15) - (3x^3 + 6x + 10)

Frequently Asked Questions

Q: What is the degree of a polynomial?
A: The degree of a polynomial is the highest power of the variable in the polynomial.

Q: Is x^(1/2) a polynomial?
A: No, because the exponent 1/2 is not a non-negative integer.

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