find the zeros and multiplicity of the equation, Range and interval q is positive.
q(x)=-x^2(x+1)^2(x-2)
since there are two double roots, the graph is tangent to the x-axis at 0 and -1. It crosses only at x=2.
So, since it is a 5th degree polynomial, with leading coefficient negative, it rises at the left end and falls on the right end.
So,
q > 0 for x < -1
q > 0 for -1 < x < 0
q > 0 for 0 < x < 2
q < 0 for x > 2
To confirm this, see the graph at
http://www.wolframalpha.com/input/?i=-x^2%28x%2B1%29^2%28x-2%29+for+-1.5+%3C+x+%3C+2.1
what are the zeros and multiplicity? and the Range of the equation?
To find the zeros and the multiplicity of the equation q(x) = -x^2(x+1)^2(x-2), we need to factorize the equation first.
The equation is already represented in factored form, where each factor represents a possible zero.
The zeros of the equation are the values of x for which q(x) equals zero. For each factor, set it equal to zero and solve for x:
1) x^2 = 0 -> x = 0 (This is a zero of multiplicity 2, as there are two instances of the factor x^2)
2) (x+1)^2 = 0 -> x + 1 = 0 -> x = -1 (This is also a zero of multiplicity 2, as there are two instances of the factor (x+1)^2)
3) x - 2 = 0 -> x = 2 (This is a simple zero, as there is only one instance of the factor (x-2))
So, the zeros of the equation q(x) are x = 0 (with multiplicity 2), x = -1 (with multiplicity 2), and x = 2 (with multiplicity 1).
To find the range and interval where q is positive, we need to analyze the signs of the factors.
- The factor x^2 is squared, so it is always positive or zero.
- The factor (x+1)^2 is squared, so it is always positive or zero.
- The factor (x-2) is negative when x < 2 and positive when x > 2.
Now, let's consider all the possibilities:
1) If x < -1, both x^2 and (x+1)^2 are positive, so the whole equation q(x) is negative.
2) If -1 < x < 0, x^2 is positive, and (x+1)^2 is positive, so the whole equation q(x) is negative.
3) If x = -1, both x^2 and (x+1)^2 are zero, so q(x) is zero.
4) If 0 < x < 2, x^2 is positive, and (x+1)^2 is positive, so the whole equation q(x) is negative.
5) If x = 2, (x-2) is zero, so q(x) is zero.
6) If x > 2, x^2 is positive, (x+1)^2 is positive, and (x-2) is positive, so the whole equation q(x) is positive.
Therefore, the range of q(x) is all positive values, and the interval where q(x) is positive is x > 2.