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.
What does the graph of a polynomial touching the x-axis at x=a tell you?
x=a is a zero with even multiplicity. The function's sign does not change at x=a.
What does the graph of a polynomial crossing the x-axis at x=a tell you?
x=a is a zero with odd multiplicity. The function's sign changes at x=a.
How can you identify the degree of a polynomial from its graph?
The degree is related to the number of turning points (local maxima and minima). A polynomial of degree n can have at most n-1 turning points. Also, consider the end behavior.
How can you identify real zeros from a polynomial's graph?
Real zeros are the x-intercepts of the graph.
What does the end behavior of a polynomial graph indicate?
The end behavior indicates the sign of the leading coefficient and whether the degree is even or odd.
How does an even function's graph look?
Symmetric about the y-axis.
How does an odd function's graph look?
Rotationally symmetric about the origin.
What does a flat region on a polynomial graph suggest?
It suggests a zero with a higher multiplicity or a turning point.
How can you determine the sign of a polynomial in different intervals from its graph?
Look at whether the graph is above (positive) or below (negative) the x-axis in each interval.
What does the y-intercept of a polynomial graph represent?
The value of the polynomial when x = 0, i.e., p(0).
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.