All categories

2 years ago

If f(1) = 0 what are all the roots of the function f(x)=x^3+3x^2-x-3 use the remainder theorem.

Mathematics

2 answers:

ArbitrLikvidat [17]2 years ago

5 0

Solution:

As we are given that f(1) = 0 .

It mean that $(x-1)$ is one of the factor of the given equation.

Remainder theorem can be applied as below:

$\frac{(x^3+3x^2-x-3)}{(x-1)}=\frac{x^3-x^2+4x^2-4x+3x-3}{(x-1)}\\ \\\frac{x^3-x^2+4x^2-4x+3x-3}{(x-1)}=\frac{x^2(x-1)+4x(x-1)+3(x-1)}{(x-1)} \\\\\frac{x^2(x-1)+4x(x-1)+3(x-1)}{(x-1)}=\frac{(x^2+4x+3)(x-1)}{(x-1)} \\\\\frac{(x^2+4x+3)(x-1)}{(x-1)} =\frac{(x^2+3x+x+3)(x-1)}{(x-1)} \\\\\frac{(x^2+3x+x+3)(x-1)}{(x-1)} =\frac{(x-1)(x+3)(x+1)}{(x-1)}$

Hence the factors are (x-1),(x+3) and (x+1).

Hence the correct option is B.

mars1129 [50]2 years ago

4 0

The answer is B quizzlet

You might be interested in

Determine whether the series is convergent or divergent. 1 2 3 4 1 8 3 16 1 32 3 64 convergent divergent Correct:

Vanyuwa [196]

Answer:

This series diverges.

Step-by-step explanation:

In order for the series to converge, i.e. $\lim_{n \to \infty} a_n =A$ it must hold that for any small $\epsilon$ >0, there must exist $n_0\in \mathbb{N}$ so that starting from that term of the series all of the following terms satisfy that $|a_n-A|n_0$ .

It is obvious that this cannot hold in our case because we have three sub-series of this observed series. One of them is a constant series with $a_n=1$ , the other is constant with $a_n=3$ , and the third one has terms that are approaching infinity.

Really, we can write this series like this:

$a_n=\begin{cases} 1 \ , \ n=4k+1, k\in \mathbb{N}_0\\ 2^{k}\ , \ n=2k, k\in \mathbb{N}_0\\3\ , \ n=4k+3, k\in \mathbb{N}_0\end{cases}$

If we denote the first series as $b_n=1$ , we will have that $\lim_{k \to \infty} b_k=1$ .

The second series is denoted as $c_k=2^k$ and we have that $\lim_{k \to \infty} c_k=+\infty$ .

The third sub-series $d_k=3$ is a constant series and it holds that $\lim_{k \to \infty} d_k=3$ .

Since those limits of sub-series are different, we can never find such $n_0\\$ so that every next term of the entire series is close to one number.

To make an example, if we observe the first sub-series if follows that A must be equal to 1. But if we chose $\epsilon =1$ , all those terms associated with the third sub-series will be out of this interval $(A-1, A+1)=(0, 2)$ .

Therefore, the observed series diverges.

5 0

2 years ago

If the shape of our data set is multimodal, we expect:

scoray [572]

Answer: (D) none of the these.

Step-by-step explanation:

A multimodal distribution refers to a distribution with two or more modes.

If the shape of our data set is multimodal, the it will show two or more peaks which represents the number modes in the data.

Since it has no relation with mean or median of the data, there for the correct option will be "none of these".

3 0

2 years ago

The first task of the Segment Two Honors Project is to select the Power Pill study participants’ groups and research board offic

Tatiana [17]

Answer:

Part 1: There are 4.7*10^21 ways to select 40 volunteers in subgroups of 10

Part 2: The research board can be chosen in 32760 ways

Step-by-step explanation:

Part 1:

The number of ways in which we can organized n elements into k groups with size n1, n2,...nk is calculate as:

$\frac{ n!}{ n1!*n2!*...*nk! }$

So, in this case we can form 4 subgroups with 10 participants each one, replacing the values of:

n by 40 participants
k by 4 groups
n1, n2, n3 and n4 by 10 participants of every subgroups

We get:

$\frac{ 40!}{10!*10!*10!*10!} = 4.7*10^{21}$

Part 2:

The number of ways in which we can choose k element for a group of n elements and the order in which they are chose matters is calculate with permutation as:

$nPk = \frac{ n!}{(n-k)!}$

So in this case there are 4 offices in the research board, those are director, assistant director, quality control analyst and correspondent. Additionally this 4 offices are going to choose from a group of 5 doctors.

Therefore, replacing values of:

n by 15 doctors
k by 4 offices

We get:

$\frac{ 15!}{ (15-4)! } = 32760$

7 0

2 years ago

Read 2 more answers

There are 8 candidates for student government: Hal, Mary, Ann, Frank, Beth, John, Emily, and Tom. The three candidates that rece

sveticcg [70]

Answer:

Step-by-step explanation:

Given that there are 8 candidates for student government: Hal, Mary, Ann, Frank, Beth, John, Emily, and Tom.

The three candidates that receive the highest number of votes become candidates for a runoff election.

i.e. 3 persons out of 8 to be selected for becoming candidates for a runoff election.

Since order does not matter we use combinations here

3 persons out of 8 can be done in 8C3 ways

= 56

no of 3-candidate combinations possible are 56

3 0

2 years ago

What are the possible polynomial expression for dimensions of the cuboid whose volume is 12y2 + 8y -20

bogdanovich [222]

Answer:

The answer is below

Step-by-step explanation:

The volume of a cuboid is the product of its length, height and breadth. It is given by:

Volume = length × breadth × height

Since the volume is given by the expression 12y² + 8y - 20. That is:

Volume = 12y² + 8y - 20 = 4(3y² + 2y - 5) = 4(3y² + 5y - 3y -5) = 4[y(3y + 5) -1(3y + 5)]

Volume = 4(y-1)(3y+5)

Volume = 12y² + 8y - 20 = 2(6y² +4y - 10) = 2(6y² + 10y - 6y -10) = 2[y(6y + 10) -1(6y + 10)]

Volume = 2(y-1)(6y+10)

Therefore the dimensions of the cuboid are either 4, y-1 and 3y+5 or 2, y-1 and 6y+10

3 0

2 years ago

Read 2 more answers