P(x)= x^5-4x^4+3x^3+2x-6 Use Descartes rule of signs to analyze the zeros
1 answer:
notice where how many "yes" sign change happen in the positive "x"
and the on the negative "-x"
for the P(x) is 3 changes, thus
so is either 3 real positive ones, or 3-2 = 1 real positive one
P(-x) gives no sign changes, so there are 0 negative ones
the polynomial has a degree of 5, thus 5 roots
if it has 3 real positive ones, then it has 2 complex ones
the complex ones pick up the slack per se
if it has 1 real positive one, then it has 4 complex ones
bear in mind complex ones always come in pairs
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Answer:
Step-by-step explanation:
This is a parabola since a quadratic is a parabola. The standard form for a parabola is y = ax² + bx + c
but before we do that, we will use the vertex form, since it will make our work easier at the beginning.
First and foremost, when we plot the vertex and the given point, the vertex is higher up than is the point; that means that this parabola opens upside down, and its vertex form will be
y=-|a|(x - h)² + k
The absolute value is out in front of the a, so we know that the value of a is positive, but the quadratic itself is negative (upside down) and we will find that math takes care of that negative that needs to be out front. So we need to solve for a by filling in the x, y, h, and k values from the point and the vertex: x = -7, y = 0, h = -3, k = 4
0 = a(-7 - (-3))² + 4 and
0 = a(-7 + 3)² + 4 and
0 = a(-4)² + 4 and
0 = a(16) + 4 and
0 = 16a + 4 and
-4 = 16a so
Now that we know a, we can plug it back into the vertex form and then put it into standard form from there.
Now we will FOIL out what's inside the parenthesis to get
Simplify by distributing the -1/4 into the parenthesis:
Combine like terms to get
And there you go!