Find the zeros for the polynomial function and give the multiplicity for each ze

Question

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around, at each zero. fx)x4x2-x 4 O 1, multiplicity 2, touches the x-axis and turns around; - 4, multiplicity 1, crosses the x-axis. O -1, multiplicity 1, crosses the x-axis 1, multiplicity 1, crosses the x-axis - 4, multiplicity 1, crosses the x-axis O -1, multiplicity 1, touches the x-axis and turns around: 1, multiplicity 1, touches the x-axis and turns around 4, multiplicity 1, touches the x-axis and turns around 4, multiplicity 1, crosses the x-axis; 1, multiplicity 1, crosses the x-axis: O 4, multiplicity 1, crosses the x-axis

Explanation / Answer

Answer: OPTION (B) -1, multiplicity 1, crosses the x-axis; 1, multiplicity 1, crosses the x-axis; -4, multiplicity 1, crosses the x-axis.

Explanation: f(x)=x3+4x2-x-4

f(x)=x3-x+4x2-4

=x(x2-1)+4(x2-1)

=(x2-1)(x+4)

f(x)=(x+1)(x-1)(x+4)       ------------------------------------------ (1)

So, we observe from eqn 1, the function value becomes zero for x=-1, 1 and -4.

Hence, the zeros of the polynomials are (-1, 1 and -4).

Now, since, all the zeros have frequencies 1 (one time values). So, all the zeros have multiplicity equal to 1.

Now, from the eqn 1, we see that

f(0)=-4, f(1)=0, f(-1)=0, f(-4)=0, f(-2)=6, f(2)=18, f(-3)=8, f(3)=56 etc. i.e.

we have points (0, -4), (1, 0) (-1, 0), (-2, 6), (2, 18), (-3, 8). (3, 56), (4, 0) etc. on the graph of f(x).

So, clearly the graph of f(x), crosses the x-axis because it is passing through the point (0, -4) which lies in the lower half of the x-axis.

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