All Flashcards
Explain the relationship between real zeros and x-intercepts.
Real zeros of a polynomial p(x) correspond to the x-intercepts of its graph, where the graph crosses the x-axis at the point (a, 0).
Explain the significance of even multiplicity.
If a zero a has an even multiplicity, the graph of p(x) touches the x-axis at x = a but does not cross it. The function's sign does not change at this zero.
How do you determine the degree of a polynomial using successive differences?
Calculate first differences, then second, third, and so on, until you find a constant difference. The number of times you need to take differences will be the degree of the polynomial.
What is the key property of even functions?
Even functions are symmetric about the y-axis and satisfy the property f(-x) = f(x).
What is the key property of odd functions?
Odd functions are rotationally symmetric about the origin and satisfy the property f(-x) = -f(x).
Describe the behavior of a graph at a zero with odd multiplicity.
The graph crosses the x-axis at the zero. The sign of the function changes at this point.
Explain the conjugate pairs theorem.
If a polynomial has real coefficients, complex zeros always come in conjugate pairs. If a + bi is a zero, then a - bi is also a zero.
What does it mean for a function to be symmetric about the y-axis?
The function is even, meaning f(-x) = f(x). The graph is a mirror image across the y-axis.
What does it mean for a function to be symmetric about the origin?
The function is odd, meaning f(-x) = -f(x). The graph looks the same when rotated 180 degrees around the origin.
How does multiplicity affect the graph's behavior at an x-intercept?
Odd multiplicity: graph crosses the x-axis. Even multiplicity: graph touches the x-axis and turns around.
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.