Overview
This lecture introduces polynomials in one variable and details the algebraic operations of addition and subtraction, focusing on identifying degree, aligning like terms, and applying key formulas.
Polynomials in One Variable
- A polynomial of degree n in one variable is written as: ( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 ).
- The coefficient ( a_n ) (of the highest degree term) must not be zero.
- Each term ( a_kx^k ) (for ( k = 0 ) to ( n )) is called a monomial.
- Polynomials can be considered as functions ( p(x) ) from real numbers to real numbers (( \mathbb{R} \to \mathbb{R} ))._
Degree and Range of Polynomial Functions
- The range of a linear polynomial (( a_1x + a_0 )) is all real numbers if ( a_1 \neq 0 ).
- Quadratic polynomials (( a_2x^2 + a_1x + a_0 )) represent parabolas, with ranges depending on the sign of ( a_2 ).
Addition of Polynomials
- To add polynomials, align like terms (same powers of ( x )) and add their coefficients.
- If a polynomial lacks a term of a certain degree, assume its coefficient for that term is 0.
- The general formula: [ (a_kx^k) + (b_kx^k) = (a_k + b_k)x^k ] for each ( k ).
- The degree of the sum is the maximum of the degrees of the two polynomials.
Examples of Addition
- Adding a degree 2 polynomial and a degree 0 polynomial results in degree 2.
- Adding polynomials of different degrees preserves the highest degree.
- For polynomials with overlapping terms, simply add the coefficients for each degree.
Algorithm for Addition of Polynomials
- Identify degrees (( m ), ( n )) of both polynomials.
- The resultant degree is ( \max(m, n) ).
- For missing degrees in a polynomial, use coefficient 0.
- Sum each pair of coefficients for like powers of ( x ).
Subtraction of Polynomials
- Subtraction is the same as addition but multiply the second polynomial's coefficients by -1.
- The formula becomes: [ (a_kx^k) - (b_kx^k) = (a_k - b_k)x^k ].
- The degree of the result is still the maximum degree of the input polynomials.
Key Terms & Definitions
- Polynomial — An expression of the form ( a_nx^n + a_{n-1}x^{n-1} + ... + a_0 ).
- Monomial — A single term ( a_kx^k ) in a polynomial.
- Degree — The highest power of ( x ) in a polynomial with a non-zero coefficient.
- Like Terms — Terms with the same exponent of ( x )._
Action Items / Next Steps
- Review and practice addition and subtraction of polynomials with varied degrees.
- Prepare for the next lecture on multiplication of polynomials.