A polynomial function has a root of negative 4 with multiplicity 4 , a root of negative 1 with multiplicity 3 , and a root of 5 with multiplicity 6 If the function has a positive leading coefficient and is of odd degree, which could be the graph of the function?
Answer
A
Image with description On a coordinate plane with x and y axes marked from negative 8 to 8, a polynomial graph is drawn. It starts from quadrant 2, passes through (negative 6 comma 8), crosses x axis at (Negative 4 comma 0), moves up and curves down at (negative 2 comma 1), crosses x axis at )negative 1 comma 0) and then y axis at (0 comma negative 1 point 9). It then moves down to quadrant 4 and curves up at (2 comma negative 4), touches x axis at (5 comma 0) and moves to quadrant 4 through (6 point 5 comma negative 8)
B
Image with description On a coordinate plane with x and y axes marked from negative 8 to 8, a polynomial graph is drawn. It starts from quadrant 3, passes through (negative 5 point 5 comma negative 8), touches x axis at (negative 4 comma 0), goes down and curves up at (negative 2 point 5 comma negative 1) and passes through x axis (negative 1 comma 0) and y axis at (0 comma 3). It moves further up and curves down at (2 comma 7), touches x axis at (5 comma 0) and moves to quadrant 1 through (6 comma 8).
C
Image with description On a coordinate plane with x and y axes marked from negative 8 to 8, a polynomial graph is drawn. It starts from quadrant 2, passes through (negative 6 comma 4), touches the x axis at (negative 4 comma 0), moves up and curves down to pass through the x axis at (negative 1 comma 0), y axis at ((0 coma negative 2). It then moves down and curves up at (3 comma negative 8), crosses the x axis at (5 comma 0) and moves to quadrant 1 passing through (5 point 5 comma 8).
D
Image with description On a coordinate plane with x and y axes marked from negative 8 to 8, a polynomial graph is drawn. It starts from quadrant 2, passes through (negative 5 point 6 comma 8), crosses x axis at (negative 4 comma 0), moves down and curves up and crosses x axis at (negative 1 comma 0). From here it passes through y axis at (0 comma 0 point 5), move down, curves up at (3 point 5 comma negative 4 point 5), crosses x axis at (5 comma 0) and move to quadrant 1 through (6 comma 8).
Explanation
The degree is 4 + 3 + 6 = 13, which is odd. With a positive leading coefficient, the graph falls to the left and rises to the right, so its ends should be in quadrants 3 and 1. At roots with even multiplicity, the graph touches the x-axis and turns; at a root with odd multiplicity, it crosses. Choice B has the correct end behavior, touches at −4 and 5, and crosses at −1.
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