9. w +w+w+w: I’m a little stuck
1 answer:
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Answer:
Explanation:
In a right triangle the angle between the hypotenuse and the adjacent leg to the angle are related by the trigonometric ratio cosine.
Cosine is the ratio of the adjacent leg to the hypotenuse:
In the right triangle ART, AT is the hypotenuse, AR is the ajacent leg to the angle 66º, and RT is the opposite leg to the angle 66º.
Then, the length of AT and 12.5 is related by the cosine ratio:
Multiplying both sides of the equation by 12.5in:
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Answer:
- a = 1
- b = -28
- P(x) = (x +2)(x -3)(x^2 +3x +10)
Step-by-step explanation:
a) Since x+2 is a factor, we know P(-2) = 0.
P(x) = (((x +2)x +a)x +b)x -60
Then the value of P(-2) is ...
P(-2) = 0 = (((0)(-2) +a)(-2) +b)(-2) -60 = (-2a +b)(-2) -60 = 4a -2b -60
We know the remainder from division by (x+3) is 60, so
P(-3) = 60 = (((-3+2)(-3) +a)(-3) +b)(-3) -60 = ((3+a)(-3) +b)(-3) -60
= (-9 -3a +b)(-3) -60 = 27 +9a -3b -60
93 = 9a -3b
These two equations can be put into standard form:
2a -b = 30
3a -b = 31
Then we have the solution ...
a = 1 . . . . . (by subtracting the first equation from the second)
-28 = b . . . by substituting into the first equation
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b) To show that (x-3) is a factor we need to evaluate P(3).
P(3) = (((3 +2)(3) +1)(3) -28)(3) -60 = (48 -28)(3) -60 = 0
The function value is 0, so (x -3) is a factor.
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c) We want to find Q(x) = x^2 +cx +d such that ...
(x +2)(x -3)Q(x) = P(x)
(x^2 -x -6)(x^2 +cx +d) = x^4 +2x^3 +x^2 -28x -60
x^4 +(c-1)x^3 +(-6-c+d)x^2 +(-6c-d)x -6d = x^4 +2x^3 +x^2 -28x -60
This gives rise to the equations ...
c -1 = 2 ⇒ c = 3
-6d = -60 ⇒ d = 10
Then P(x) can be factored as ...
P(x) = (x +2)(x -3)(x^2 +3x +10)
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<em>Comment on the attached graph</em>
I like to use a graphing calculator to find real roots of higher-degree polynomials. This graph shows the real zeros to be -2 and +3, so we know that (x +2) and (x -3) are factors. The green curve is P(x) with those factors divided out, so is a graph of Q(x). The vertex of that graph tells us that ...
Q(x) = (x +1.5)^2 +7.75 = x^2 +3x +10
