Notes: Lecture on Polynomials

Subject Overview

• This lecture explains about polynomials.
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Main Points

Definition of Polynomials

• A polynomial consists of variables and constants.
• The powers of the variables should always be whole numbers.
• Examples of polynomials: 2X + 3, X^2 + 2X + 1
• Examples of non-polynomials: X^-1, 5X^2.5

Degree of Polynomials

• The degree of a polynomial is the highest power of its variable.
• Example: The degree of X^3 + 4X^2 + X + 5 is 3.
• Examples of finding degrees are also provided.

Monomials and Terms

• Monomials mean single-term polynomials.
• Terms are separated by plus or minus signs.
• Multiplication and division of variables are not considered separate terms.

General Form of Polynomials

• Polynomials are generally written in descending order.
• General form: an*x^n + an-1*x^(n-1) + ... + a1*x + a0

Zero of Polynomials

• The zero of a polynomial is the value that makes the polynomial equal to 0.
• Example: The zero of 2X - 4 = 0 is X = 2.
• Examples and questions for checking zeros are also given.

Remainder Theorem

• The Remainder Theorem states that if P(x) is divided by (x - a), the remainder will be P(a).
• A trick for directly finding the remainder is also given. Example: The value obtained by substituting x = 3 in P(x) will be the remainder.

Divide and Remainder

• Finding the remainder by dividing larger polynomials by smaller polynomials.
• Example: In P(x) ÷ D(x), if D(x) is zero, the remainder is P(a).
• Factorization and practice questions are given.

Hints and Tips

• Pay special attention to brackets when finding zeros and remainders.
• Practice different approaches for different types of polynomials.
• Carefully consider all types of terms and factorize them correctly.

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In Conclusion

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