Summer Algebra 2 Flashcards

Questions and Answers

What is the definition of complex numbers?

a+bi

a+bi, a-bi (correct)

a-bi

a-b

What are the conjugates of 2-5i?

2+5i

What is a double root?

An equation with two identical factors, e.g., (x-7)^2

What does the Remainder Theorem state?

The remainder on dividing P(x) by x-c is P(c) Signup and view all the answers

If and only if ______, then _ is a factor of _ and ___ is a root of P(x)=0.

P(c)=0, x-c, p(x), c Signup and view all the answers

What does the Fundamental Theorem of Algebra state?

Every polynomial of degree n has exactly n zeros Signup and view all the answers

What does the Conjugate Pairs Theorem state?

If a+bi is a zero, so is a-bi Signup and view all the answers

What does the Rational Zeros Theorem define?

If the coefficient before x^n is 1, then the roots are factors of the last number without x in the polynomial Signup and view all the answers

What is the significance of c when a polynomial has a factor (x-c)^n?

c is a zero of multiplicity n Signup and view all the answers

What does Descartes' Rule of Signs provide information about?

Number and location of the real roots of a polynomial equation Signup and view all the answers

What is the number of positive real roots equal to?

Number of variations in sign Signup and view all the answers

How is the number of negative real roots determined?

Number of variations in sign of P(-x)=0 Signup and view all the answers

What are imaginary roots?

Any roots left over that are neither positive nor negative Signup and view all the answers

Why is the Theorem on Bounds important?

It restricts the possible roots that lie between the upper and lower bounds Signup and view all the answers

If r is positive and the numbers in the last row of synthetic division are positive, what can be said about r?

r is an upper bound Signup and view all the answers

What does it mean if no root is greater than r?

Upper bound Signup and view all the answers

If r is negative and the numbers in the last row of synthetic division are alternatively positive and negative, what does this imply?

Lower bound - no root less than r Signup and view all the answers

What defines an even function?

f(-x) = f(x) Signup and view all the answers

Even functions are symmetric about which axis?

y-axis Signup and view all the answers

What defines an odd function?

f(-x) = -f(x) Signup and view all the answers

Odd functions are symmetric about which point?

Origin Signup and view all the answers

If the largest exponent in y=f(x) is even, what happens to the two ends of the graph?

They point in the same direction Signup and view all the answers

If all exponents in y=f(x) are even numbers, how is the graph positioned?

Symmetric about the y-axis Signup and view all the answers

If all exponents in y=f(x) are odd numbers and there is no constant term, what can be said about the graph?

Symmetric about the origin Signup and view all the answers

In the complex number system, what does a polynomial function of degree n greater than or equal to 1 have?

At least one zero Signup and view all the answers

What is the iterative process used for?

Estimating irrational zeros Signup and view all the answers

What does the Intermediate Value Theorem state?

If a continuous function takes on opposite signs at two points, then there exists at least one c between them such that f(c) = 0 Signup and view all the answers

Study Notes

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 ).

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Test your knowledge of key concepts in Algebra 2 with these flashcards. Explore definitions and examples of complex numbers, conjugates, double roots, and more. Perfect for summer revision or quick study sessions.