1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Flura [38]
2 years ago
11

Part 3

Mathematics
1 answer:
Luden [163]2 years ago
3 0

Answer:

   possible rational roots: ±1/2, ±1, ±5/2, ±5, ±25/2, ±25

   p(x) = (x +1)^2(2x -5)(x^2 -4x +5)

   complex roots: 2±i

   see the last attachment for a graph

Step-by-step explanation:

2. The leading coefficient of p(x) is 2, and the constant term is 25. The Rational Root Theorem tells you possible rational roots will be of the form ...

  ±(divisor of 25)/(divisor of 2)

That is, they are ...

  ±1/2, ±1, ±5/2, ±5, ±25/2, ±25

__

3. Before we get into synthetic division, we choose to see if we can reduce this list any. We note that p(0) = -25. The value of p(1) is the sum of the coefficients:

  p(1) = 2 -9 +6 +22 -20 -25 = 30 -54 = -24

Similarly, the value of p(-1) is the same sum with odd-degree coefficients negated:

  p(-1) = -2 -9 -6 +22 +20 -25 = 42 -42 = 0

So, we found our first root: -1. Using synthetic division, we can reduce the polynomial and start over. See the first attachment for this division.

__

The reduced polynomial is ...

  p1(x) = 2x^4 -11x^3 +17x^2 +5x -25

We already know that +1 is not a of it. Checking -1, we have ...

  p1(-1) = 2 +11 +17 -5 -25 = 0

So, we found our second root: -1. Using synthetic division, we can reduce the polynomial and start over. See the second attachment for this division.

__

The reduced polynomial is ...

  p2(x) = 2x^3 -13x^2 +30x -25

The alternating signs tell us there are no more negative real roots. They also tell us there are 1 or 3 positive real roots. We know p2(0) = -25. Then ...

  p2(1) = 2 -13 +30 -25 = 32 -38 = -6

The average rate of change between these points is (-6 -(-25))/(1 -0) = 19. At this rate, we expect a root between x=1 and x=2. Testing x=2 using synthetic division, we get a remainder of -1. (See the 3rd attachment.) Then the rate of change between x=1 and x=2 is (-1 -(-6))/(2-1) = 5, suggesting x=5/2 might be a worthwhile test value.

The synthetic division is shown in the 4th attachment. You will note that we divide the polynomial p2(x) by its leading coefficient, so the coefficients used for p2(x) in the synthetic division are 1, -13/2, 15, -25/2. The remainder of 0 tells us that (x -5/2) is a factor of p2(x)/2, or (2x -5) is a factor of p2(x).

__

The reduced polynomial is ...

  p3(x) = x^2 -4x +5

This can be written in vertex form as ...

  p3(x) = (x -2)^2 +1

The positive leading coefficient means the graph opens upward, and the vertex at (2, 1) means there are no real solutions.

The real solutions to p(x) are x = -1, -1, and 5/2.

__

4. The complex solutions will be the solutions to ...

  (x -2)^2 +1 = 0

  (x -2)^2 = -1

  x -2 = ±√(-1) = ±i

  x = 2 ±i . . . . complex roots of p(x)

__

5. The graph is shown in the last attachment. The odd degree and positive leading coefficient of p(x) means the overall shape will be from lower left to upper right (/). That is, the sign of the end value of p(x) will match the sign of x.

The graph will touch the x-axis from below at x = -1, and will cross at x = 2.5. There is no particular symmetry.

The final quadratic factor is graphed and its vertex shown. The vertex matches that of the vertex-form equation for p3(x), above.

You might be interested in
N - 4 = 3n + 6<br> what number does n represent?
pav-90 [236]

Answer:

n = -5

Step-by-step explanation:

n - 4 = 3n + 6

n = 3n + 10

-2n = 10

n = -5

7 0
3 years ago
Read 2 more answers
List the common multiple of 42 and 14 . Help plzzzz 
Alborosie
The answer is 42,84,126 and so on.(you keep on adding 42)
3 0
3 years ago
<img src="https://tex.z-dn.net/?f=x%20%3D%20%20%5Cfrac%7Bb%20%2B%20%20-%20%20%5Csqrt%7B%7Bb%7D%5E%7B2%7D%20-%204ac%20%7D%20%7D%7
Phantasy [73]

If it is asking if that equation is the quadratic formula, then the answer is false. The reason why is that the first 'b' should be negative

The quadratic formula is

x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

3 0
3 years ago
A bag of marbles contains 16 blue, 8 green, 9 red, 12 yellow and 5 black marbles. What is the probability of drawing a green mar
Novay_Z [31]

Answer:

Step-by-step explanation:

4 0
3 years ago
Use the maclaurin series for cos(x) to find a maclaurin series for sin^2(x)
kolbaska11 [484]

Try this solution (see the attachment).

5 0
3 years ago
Other questions:
  • The graph of the function f(x) = (x + 6)(x + 2) is shown.
    10·2 answers
  • Paul bought a 15-year treasury bond for a face amount of $600. The 2.5% interest will be compounded quarterly. What will the fut
    15·2 answers
  • What is the equation of a line that passes through (1, 3) and has a slope of 2?
    11·1 answer
  • Solve the system of equations using any appropriate method: 5x + 2y = -4 and -x + 2y = 8
    11·1 answer
  • The product of two consecutive integers is 56.find the the integers.
    13·1 answer
  • Whats the answer to this question ?
    5·2 answers
  • HELP please and HURRY
    11·1 answer
  • What is the value of a in the equation 38+ B equals 54 would b equals nine
    10·1 answer
  • PLEASE HELP ME I WILL MARK BRAINLIEST
    11·2 answers
  • (x4 - 4x3 + 2x2 - 4x + 1) / (x2 + 1)
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!