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Question
Classify the following as a constant, linear, quadratic and cubic polynomials:
1 + x + x2
Solution
Constant polynomials: The polynomial of the degree zero.
Linear polynomials: The polynomial of degree one.
Quadratic polynomials: The polynomial of degree two.
Cubic polynomials: The polynomial of degree three.
Powers of x = 1 and 2, respectively.
The highest power of the variable x in the given expression = 2
Hence, the degree of the polynomial = 2
Since it is a polynomial of degree 2, it is a quadratic polynomial.
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Case Study -1
The figure given alongside shows the path of a diver, when she takes a jump from the diving board. Clearly it is a parabola.
Annie was standing on a diving board, 48 feet above the water level. She took a dive into the pool. Her height (in feet) above the water level at any time‘t’ in seconds is given by the polynomial h(t) such that h(t) = -16t2 + 8t + k.
What is the value of k?
The below picture are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.
The graph of x2 + 1 = 0
