1.number -4 is a solution to the equation because the remainder of the division

1.number -4 is a solution to the equation because the remainder of the division, x^(3)+18x^(2)+101x+180 divided by x+4

2.iris has 120 songs on her playlist. each week, she downloads 4 new song

3.X ~ N(0,1); P(X > .25 or X <>

The correct answer and explanation is:

Let’s go through each statement and explain them.

1. Remainder of Polynomial Division:

The first statement suggests that −4-4 is a solution to the equation because the remainder of dividing the polynomial x3+18×2+101x+180x^3 + 18x^2 + 101x + 180 by x+4x + 4 is zero.

• Explanation: When you divide a polynomial by a binomial of the form x−ax – a, the remainder theorem states that the remainder is equal to the value of the polynomial evaluated at x=ax = a. In this case, you are dividing by x+4x + 4, which is the same as x−(−4)x – (-4). Therefore, to check if −4-4 is a root of the polynomial, substitute x=−4x = -4 into the polynomial x3+18×2+101x+180x^3 + 18x^2 + 101x + 180. Let’s substitute: P(−4)=(−4)3+18(−4)2+101(−4)+180P(-4) = (-4)^3 + 18(-4)^2 + 101(-4) + 180 =−64+18(16)−404+180= -64 + 18(16) – 404 + 180 =−64+288−404+180= -64 + 288 – 404 + 180 =0= 0 Since P(−4)=0P(-4) = 0, the remainder is zero, confirming that −4-4 is indeed a solution to the equation.

2. Iris’s Playlist:

The second statement says Iris has 120 songs on her playlist, and each week she downloads 4 new songs.

• Explanation: To find out how many songs Iris will have after a certain number of weeks, we can use a simple linear model. Let the number of weeks be nn, and the total number of songs after nn weeks be S(n)S(n). The total number of songs increases by 4 songs each week, starting from 120 songs. The formula for the total number of songs after nn weeks is: S(n)=120+4nS(n) = 120 + 4n If we wanted to know how many songs she has after 5 weeks, for example, we would substitute n=5n = 5: S(5)=120+4(5)=120+20=140S(5) = 120 + 4(5) = 120 + 20 = 140 So, after 5 weeks, Iris will have 140 songs.

3. Probability Statement:

The third statement involves a probability question: X∼N(0,1)X \sim N(0,1), and we are asked to find P(X>0.25 or X<−0.25)P(X > 0.25 \text{ or } X < -0.25).

• Explanation: The symbol X∼N(0,1)X \sim N(0, 1) means that XX follows a standard normal distribution, where the mean is 0 and the standard deviation is 1. The probability expression asks for the likelihood that XX is either greater than 0.25 or less than -0.25. Using symmetry of the standard normal distribution:
  • P(X>0.25)P(X > 0.25) is the probability that XX is greater than 0.25.
  • P(X<−0.25)P(X < -0.25) is the probability that XX is less than -0.25.
  The probability for these two events can be found by looking up the corresponding z-scores in a standard normal distribution table, or using a calculator: P(X>0.25)=1−P(X≤0.25)≈1−0.5987=0.4013P(X > 0.25) = 1 – P(X \leq 0.25) \approx 1 – 0.5987 = 0.4013 P(X<−0.25)=P(X≤−0.25)≈0.4013P(X < -0.25) = P(X \leq -0.25) \approx 0.4013 Since the two events are mutually exclusive (i.e., they cannot both occur at the same time), the total probability is the sum of these two probabilities: P(X>0.25 or X<−0.25)=0.4013+0.4013=0.8026P(X > 0.25 \text{ or } X < -0.25) = 0.4013 + 0.4013 = 0.8026 Therefore, the probability is approximately 0.8026.

In summary:

• The remainder theorem confirms that −4-4 is a solution.
• Iris’s playlist grows linearly by 4 songs per week.
• The probability of XX being greater than 0.25 or less than -0.25 is approximately 0.8026.