All Flashcards
Define a real number.
A number that can be found on a number line (integers, fractions, decimals, irrational numbers).
Define an imaginary number.
A number involving the imaginary unit i, where , in the form bi, where b is a real number.
Define a complex number.
A number in the form a + bi, where a and b are real numbers.
Define a zero of a polynomial.
A value a such that p(a) = 0.
Define a linear factor.
If a is a zero of p(x), then (x - a) is a linear factor of p(x).
Define multiplicity of a zero.
The number of times a linear factor appears in the factored form of a polynomial.
Define x-intercept.
The point (a,0) where the graph of p(x) crosses or touches the x-axis, where 'a' is a real zero.
Define an even function.
A function that is symmetric about the y-axis, satisfying the property f(-x) = f(x).
Define an odd function.
A function that is rotationally symmetric about the origin, satisfying the property f(-x) = -f(x).
Define the degree of a polynomial.
The highest power of x in the polynomial.
How do you find all zeros (real and complex) of a polynomial?
- Use the Rational Root Theorem to find potential rational roots. 2. Use synthetic division to test the potential roots. 3. Factor the polynomial completely. 4. Solve for the remaining zeros (may involve the quadratic formula).
How do you determine if a function is even or odd algebraically?
- Find f(-x). 2. If f(-x) = f(x), the function is even. 3. If f(-x) = -f(x), the function is odd. 4. If neither, the function is neither even nor odd.
How do you sketch a polynomial graph given its zeros and multiplicities?
- Plot the x-intercepts (zeros). 2. Determine the end behavior based on the leading term. 3. Consider the behavior at each x-intercept based on multiplicity (cross or touch). 4. Sketch a smooth curve connecting the intercepts.
How do you determine the intervals where a polynomial is positive or negative?
- Find all real zeros. 2. Create a number line with the zeros. 3. Test a value in each interval. 4. Determine the sign of the polynomial in each interval.
How do you find a polynomial given its zeros and their multiplicities?
- Write the linear factors corresponding to each zero. 2. Raise each factor to the power of its multiplicity. 3. Multiply the factors together. 4. Multiply by a constant 'a' if another point on the polynomial is given.
How do you use synthetic division to find zeros?
- Write down the coefficients of the polynomial and the potential root. 2. Bring down the first coefficient. 3. Multiply by the root and add to the next coefficient. 4. Repeat until the last coefficient. 5. If the remainder is 0, the root is a zero.
How do you factor a polynomial by grouping?
- Group terms in pairs. 2. Factor out the greatest common factor from each pair. 3. If the binomial factors are the same, factor out the binomial. 4. Simplify.
How do you use the quadratic formula to find complex zeros?
- Identify a, b, and c in the quadratic equation . 2. Substitute into the quadratic formula: . 3. Simplify. 4. If the discriminant () is negative, the roots are complex.
How do you determine the end behavior of a polynomial?
- Identify the leading term (). 2. If n is even and a > 0, both ends go up. 3. If n is even and a < 0, both ends go down. 4. If n is odd and a > 0, left end goes down, right end goes up. 5. If n is odd and a < 0, left end goes up, right end goes down.
How do you determine the degree of a polynomial from a table of values?
- Calculate successive differences (first, second, third, etc.). 2. The degree is the number of times you need to take differences until you get a constant value.
What are the differences between real and imaginary numbers?
Real Numbers: Can be plotted on a number line, no imaginary component | Imaginary Numbers: Involve the imaginary unit i = sqrt(-1), cannot be plotted on a standard number line
What are the differences between even and odd functions?
Even Functions: Symmetric about the y-axis, f(-x) = f(x) | Odd Functions: Symmetric about the origin, f(-x) = -f(x)
What are the differences between zeros with even and odd multiplicities?
Even Multiplicity: Graph touches the x-axis and turns around, sign of f(x) does not change | Odd Multiplicity: Graph crosses the x-axis, sign of f(x) changes
Compare and contrast x-intercepts and y-intercepts.
X-intercepts: Points where the graph crosses the x-axis, y=0, represent real zeros | Y-intercepts: Point where the graph crosses the y-axis, x=0, represents the value of the function at x=0
Compare and contrast polynomials with real coefficients and polynomials with complex coefficients.
Real Coefficients: Complex roots occur in conjugate pairs | Complex Coefficients: Complex roots do not necessarily occur in conjugate pairs
What is the difference between a root and an x-intercept?
Root: A solution to the polynomial equation p(x) = 0, can be real or complex | X-intercept: A point where the graph crosses the x-axis, only represents real roots
Compare and contrast the graphs of even and odd degree polynomials.
Even Degree: Ends go in the same direction (both up or both down) | Odd Degree: Ends go in opposite directions (one up, one down)
Compare and contrast linear and quadratic functions.
Linear: Degree 1, graph is a straight line | Quadratic: Degree 2, graph is a parabola
Compare and contrast rational and irrational numbers.
Rational: Can be expressed as a fraction p/q where p and q are integers | Irrational: Cannot be expressed as a fraction, decimal representation is non-repeating and non-terminating
Compare and contrast multiplicity 1 and multiplicity 2.
Multiplicity 1: The graph crosses the x-axis | Multiplicity 2: The graph touches the x-axis and turns around.