Summer Algebra 2 Flashcards

Complex Numbers and Conjugates

• Complex numbers take the form ( a + bi ) where ( a ) and ( b ) are real numbers and ( i ) is the imaginary unit.
• The conjugate of a complex number ( a + bi ) is ( a - bi ), and vice versa.

Roots and Factors

• A double root occurs when an equation has two identical factors, exemplified by ( (x - 7)^2 ).
• The Remainder Theorem states that the remainder from dividing ( P(x) ) by ( x - c ) is equal to ( P(c) ).
• According to the Factor Theorem, ( P(c) = 0 ) indicates that ( x - c ) is a factor of ( P(x) ) and ( c ) is a root.

Polynomial Theorems

• The Fundamental Theorem of Algebra asserts every polynomial of degree ( n ) has exactly ( n ) zeros.
• The Conjugate Pairs Theorem states that if ( a + bi ) is a zero of a polynomial, then ( a - bi ) is also a zero.

Zeros and Roots

• The Rational Zeros Theorem suggests that if the coefficient of ( x^n ) is ( 1 ), then the roots are factors of the constant term in the polynomial.
• A polynomial with a factor ( (x - c)^n ) indicates ( c ) has a multiplicity of ( n ) as a zero.

Root Analysis

• Descartes' Rule of Signs helps determine the number and location of real roots of a polynomial equation.
• The number of positive real roots corresponds to the number of sign variations in the polynomial function.
• The number of negative real roots is found by examining the variations in sign for ( P(-x) ).

Root Behavior and Boundaries

• Imaginary roots refer to any remaining roots that are not purely positive or negative.
• The Theorem on Bounds helps identify potential roots by setting upper and lower limits.
• If ( r ) is a positive number and all values in synthetic division's last row are positive, then ( r ) is confirmed as an upper bound.

Function Symmetry and Behavior

• Even functions satisfy the equation ( f(-x) = f(x) ) and show symmetry about the y-axis.
• Odd functions meet the condition ( f(-x) = -f(x) ) and exhibit symmetry about the origin.
• When the highest exponent in ( y = f(x) ) is even, the graph's ends point in the same direction.

Graphical Attributes

• If all exponents in ( y = f(x) ) are even, the function is symmetric regarding the y-axis.
• A function with all odd exponents and no constant term is symmetric about the origin.

Complex Polynomials

• In the complex number system, any polynomial function of degree ( n ) (where ( n \geq 1 )) will have at least one zero.

Estimation Processes

• The iterative process is a method used for estimating irrational zeros of polynomials.
• The Intermediate Value Theorem states that for any value ( a ) between ( f(x_1) ) and ( f(x_2) ), there exists at least one ( c ) in the interval where ( f(c) = a ).