One of the zeros of a certain quadratic polynomial with real coefficients is 1 +
ID: 3006181 • Letter: O
Question
One of the zeros of a certain quadratic polynomial with real coefficients is 1 + i. What is its other zero? 38. The graph of a certain cubic polynomial function, f, has one x-intercept at (1,0) that crosses the x-axis, and another x-intercept at (-3,0) that touches the x-axis but does not cross it. What are the zeros of f and their multiplicities? 39. Explain why there cannot be two different points at which the graph of a cubic polynomial touches the x-axis without crossing it. 40. Why can't the numbers i, 2i, 1, 2 be the set of zeros of some fourth-degree polynomial with real coefficients? 41. The graph of a polynomial function is given below. What is the lowest possible degree of this polynomial? Explain. Find a possible expression for the function.Explanation / Answer
The limiting behavior of a cubic(or any ) function describes what happens to the function as x ±.
The degree of a polynomial and the sign of its leading coefficient dictates its limiting behavior.
For Cubic Function (or any odd function), the limiting behaviour is as follows:
1. If f(x) is an odd degree polynomial with positive leading coefficient, thenf(x) - as x - and f(x) as x .
2.If f(x) is an odd degree polynomial with negative leading coefficient, thenf(x) as x - and f(x) - as x .
From above it is clear that , graph crosses x.axis.
To explain clearly, we need to know about roots and Turning points.
For nth degree polynomial can have at most n real roots (x-intercepts or zeros) counting multiplicities. And nth degree polynomial need not have n real roots — it could have less because it has imaginary roots. Notice that an odd degree polynomial must have at least one real root since the function approaches - at one end and
+ at the other...
In addition, an nth degree polynomial can have at most n - 1 turning points. A turning point is a point at which the function changes from increasing to decreasing or decreasing to increasing as seen in the figure below. Again, an nth degree polynomial need not have n - 1 turning points, it could have less.
Cubic functions can have at most 3 real roots (including multiplicities) and 2 turning points.
The multiplicity of a root affects the shape of the graph of a polynomial. Specifically,
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