All categories

3 years ago

List all possible rational roots. Then use synthetic division to confirm which rational roots exist:

Mathematics

1 answer:

Kisachek [45]3 years ago

6 0

Answer:

$\boxed{(1) \, x = \, \pm \dfrac{1}{2}, \pm 1, \pm2, \pm \dfrac{5}{2}, \pm 5, \pm 10; (2) \, x = -2}$

Step-by-step explanation:

2x³+ 6x² - x - 10 = 0

(1) Possible roots

The Rational Roots Theorem states that, if a polynomial has any rational roots, they will have the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

$\text{Possible rational root} = \dfrac{ p }{ q } = \dfrac{\text{factor of constant term}}{\text{factor of leading coefficient}}$

In your function, the constant term is -10 and the leading coefficient is 2, so

$\text{Possible root} = \dfrac{\text{factor of 10}}{\text{factor of 2}}$

Factors of 10 = ±1, ±2, ±5, ±10

Factors of 2 = ±1, ±2

$\text{Possible roots are } \large \boxed{\mathbf{x = \pm \dfrac{1}{2}, \pm 1, \pm2, \pm \dfrac{5}{2}, \pm 5, \pm 10}}$

(2) Synthetic division

Rather than work through all 12 possibilities, I will do one that works.

$\begin{array}{r|rrrr}-2 & 2 & 6 & -1 & -10\\& & -4& -4 & 10\\& 2 & 2& -5 & 0\\\end{array}$

So, x = -2 is a root, and the quotient is 2x² + 2x - 5.

(3) Check for other rational roots

2x² + 2x - 5 = 0

D = b² - 4ac =2²- 4(2)(-5) = 4 + 40 = 44

√44 = 2√11, which is irrational.

Since irrational roots come in pairs, the cubic equation has two real, irrational roots and one rational root at x = -2.

You might be interested in

According to the 2010 census data, the population of Texas was about 25,000,000 people. The land area of Texas is about 260,000

Monica [59]

Answer:249037470.178 peoples per mi^2

8 0

1 year ago

Read 2 more answers

Given: △ABC, m∠A=60°, m∠C=45°, AB=9 Find: Perimeter of △ABC, Area of △ABC

stellarik [79]

Answer:

Perimeter of ΔABC: $\frac{27}{2}$ + $\frac{9}{2} * \sqrt6$ units

Area of ΔABC: $\frac{81}{8}*\sqrt3 + \frac{243}{8}$ units

Skills required: HS Geo, Special Triangles

Step-by-step explanation:

1) The best option is to break down this triangle. Let's draw an altitude from Point B down to Segment AC. The point from the altitude that intersects AC is Point D. BD is the height of our triangle, AC is the base.

2) Angle A is 60 degrees, and since Angle BDA is 90 degrees, Angle ABD is 30 degrees. We can use the 30-60-90 degree right triangle property for the triangle BDA.

This states that if the side opposite the 30 degree angle is $x$ , the side opposite the 60 degree angle is $x*\sqrt3$ , and the side opposite the 90 degree angle is $2x$ .

AB is 9 units, and it is opposite the 90 degree angle. This means that $2x=9, x = \frac{9}{2}$ ==> This then means that AD, the segment opposite the 30 degree angle in this triangle is $\frac{9}{2}$ units. Segment BD (the height) is $\frac{9}{2} * \sqrt3$ .

3) Angle C is 45 degrees, and Angle BDC is 90 degrees, which means that Angle CBD is 45 degrees. We can use the 45-45-90 degree right triangle property for the triangle BCD.

This states that if the side opposite the 45 degree angle is $x$ , the other side opposite a 45 degree angle is also $x$ , but the hypotenuse (side opposite the right (90 degree) angle) is $\sqrt{2}*x$ .

BD is $\frac{9}{2} * \sqrt3$ , which means DC is the same. BC, which is the hypotenuse is BD multiplied by square-root-2, which is $\frac{9}{2} * \sqrt6$ .

4) Area is $\frac{1}{2}*b*h$ , the base (b) is AC (which is $\frac{9}{2}+\frac{9}{2}*\sqrt3$ ), the height is BD ( $\frac{9}{2}*\sqrt3$ ). When multiple you will get $\frac{81}{4}*\sqrt3 + \frac{243}{4}$ , then this multiplied by 1/2 is

$\frac{81}{8}*\sqrt3 + \frac{243}{8}$ <--> this is the area!

5) Perimeter is just the sum of all side: 9 + $\frac{9}{2}$ + $\frac{9}{2} * \sqrt6$ = $\frac{27}{2}$ + $\frac{9}{2} * \sqrt6$ unit

6 0

2 years ago

One environmental group did a study of recycling habits in a California community. It found that 75% of the aluminum cans sold i

AysviL [449]

Answer:

$P( \^ p > 0.775 ) = 0.12798$

$P( 0.6718 < p < 0.775 ) =0.87183$

Step-by-step explanation:

From the question we are told that

The population proportion is $p = 0.75$

Considering question a

The sample size is $n = 387$

Generally the standard deviation of this sampling distribution is

$\sigma = \sqrt{ \frac{p(1 - p)}{ n } }$

=> $\sigma = \sqrt{ \frac{0.75(1 - 0.75)}{ 387 } }$

=> $\sigma = 0.022$

The sample proportion of cans that are recycled is

$\^ p = \frac{ 300}{387 }$

=> $\^ p = 0.775$

Generally the probability that 300 or more will be recycled is mathematically represented as

$P( \^ p > 0.775 ) = P( \frac{\^ p - p }{ \sigma } > \frac{0.775 - 0.75 }{ 0.022} )$

$\frac{\^ p - p }{\sigma } = Z (The \ standardized \ value\ of \ \^ p )$

$P( \^ p > 0.775 ) = P( Z > 1.136 )$

From the z table the area under the normal curve to the left corresponding to 1.591 is

$P( Z > 1.136) = 0.12798$

=> $P( \^ p > 0.775 ) = 0.12798$

Considering question b

Generally the lower limit of sample proportion of cans that are recycled is

$\^ p_1 = \frac{ 260 }{387 }$

=> $\^ p_1 = 0.6718$

Generally the upper limit of sample proportion of cans that are recycled is

$\^ p_2 = \frac{ 300}{387 }$

=> $\^ p_2 = 0.775$

Generally probability that between 260 and 300 will be recycled is mathematically represented as

$P( 0.6718 < p < 0.775 ) = P( \frac{0.6718 - 0.75 }{ 0.022}< \frac{\^ p - p }{ \sigma }$

=> $P( 0.6718 < p < 0.775 ) = P( -3.55 < Z < 1.136 )$

=> $P( 0.6718 < p < 0.775 ) = P(Z < 1.136 ) - P( Z < -3.55 )$

From the z table the area under the normal curve to the left corresponding to 1.136 and -3.55 is

$P( Z < -3.55 ) = 0.00019262$

and

$P(Z < 1.136 ) = 0.87202$

$P( 0.6718 < p < 0.775 ) = 0.87202- 0.00019262$

=> $P( 0.6718 < p < 0.775 ) =0.87183$

4 0

2 years ago

Suppose has a solution. Explain why the solution is unique precisely when has only the trivial solution. Choose the correct answ

Margarita [4]

Complete question is;

Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0 has only the trivial solution. Choose the correct answer.

A. Since Ax = b is inconsistent, its solution set is obtained by translating the solution set of Ax = 0. For Ax = b to be inconsistent, Ax = 0 has only the trivial solution.

B. Since Ax = b is consistent, its solution set is obtained by translating the solution set of Ax = 0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax = 0 is a single vector, and that happens if and only if Ax = 0 has only the trivial solution.

C. Since Ax = b is inconsistent, then the solution set of Ax = 0 is also inconsistent. The solution set of Ax = 0 is inconsistent if and only if Ax = 0 has only the trivial solution.

D. Since Ax = b is consistent, then the solution is unique if and only if there is at least one free variable in the corresponding system of equations. This happens if and only if the equation Ax = 0 has only the trivial solution.

Answer:

Option B: Since Ax = b is consistent, its solution set is obtained by translating the solution set of Ax = 0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax = 0 is a single vector, and that happens if and only if Ax = 0 has only the trivial solution

Step-by-step explanation:

There are different ways of explaining this but we will explain it algebraic ally.

If Ax = b has a solution, then it can be said to be unique if and only if every column of A will be a pivot column. Now, If every column of A will be a pivot column, then it means that there are no free variables, and thus the homogeneous equation will have only the trivial solution.

Also homogeneous equations are always constant.

Thus the correct option is Option B: Since Ax = b is consistent, its solution set is obtained by translating the solution set of Ax = 0. So the solution set of Ax = b is a single vector if and only if the solution set of Ax = 0 is a single vector, and that happens if and only if Ax = 0 has only the trivial solution

5 0

3 years ago

Find cscθ. pls answer

ExtremeBDS [4]

Answer:

The answer to your question is csc Ф = $\frac{-5}{4}$

Step-by-step explanation:

Process

1.- Determine the sign

We must determine csc Ф in the forth quadrangle, here csc is negative.

2.- Determine the hypotenuse

c² = a² + b²

c² = 3² + (-4)²

c² = 9 + 16

c² = 25

c = 5

3.- Determine csc Ф

csc Ф = $\frac{hypotenuse}{opposite side}$

csc Ф = $\frac{5}{-4}$ = $\frac{-5}{4}$

7 0

2 years ago