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]
3 years ago
11

Part 3

Mathematics
1 answer:
Luden [163]3 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
The expression 2(l + w) is used to calculate the perimeter of a rectangle, where l is length and w is width. If the length is Fr
bulgar [2K]
E E E E E? If I’m right please let me know
5 0
3 years ago
The area of a rectangle is 12x^2 - 29x + 15. The width is 4x-3. What is the length of the rectangle.
allochka39001 [22]

Answer:

Step by step explanation :

Given:

it is given that in a rectangle the area is is 12x^2 -29x + 15

and the width is 4x- 3

Solution :

The rest is given in pic.

Thank u, please please mark it is Brainliest!

Hope u submit on time!

5 0
3 years ago
I NEED HELP ASAP I WILL GIVE BRAINLIST
posledela

519.99(.0825) = 42.899

519.99 + 42.899 = $562.89

3 0
3 years ago
Read 2 more answers
Write the parts of the triangle XYZ that are congruent to<br> triangle PQR
timofeeve [1]

Answer:

PQ=xy ( side)

PR=xz. (side)

QR=yz. (side)

7 0
1 year ago
Read 2 more answers
I need help pls if you can’t see let me know
NNADVOKAT [17]

Answer:

I cant see it

Step-by-step explanation:

4 0
3 years ago
Other questions:
  • Find the inverse of the following f(x)=-3x^2+3 X&gt;0
    13·1 answer
  • how many groups of 24 can we make with 324 foil beads? foil beads: 324 beads ÷ 24 beads per pair of glasses
    5·1 answer
  • Micah has the coins shown below.
    13·2 answers
  • Write the decimals as words
    5·1 answer
  • Allison spent a total of $16.20 for lunch including tax and a tip. She paid 8% sales tax on her purchase and then left a tip equ
    7·1 answer
  • Can someone help me please? I forgot how to do this
    9·2 answers
  • What is the next fraction in this sequence? Simplify your answer.<br> 1/4, 3/8, 1/2, 5/8...
    7·1 answer
  • Answer the questions. All help needed.
    6·1 answer
  • Due in 1 hour <br> screenshot below
    5·1 answer
  • A well-mixed cookie dough will produce cookies with a mean of 6 chocolate chips apiece. What is the probability of getting a coo
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!