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3 years ago

Huh.. can someone please help me, i honestly really need this rn.. :(

Mathematics

1 answer:

Harman [31]3 years ago

7 0

Answer:

€

p(x) is a polynomial, the solutions to the equation

€

p(x) = 0 are called the zeros of the

polynomial. Sometimes the zeros of a polynomial can be determined by factoring or by using the

Quadratic Formula, but frequently the zeros must be approximated. The real zeros of a polynomial

p(x) are the x-intercepts of the graph of

€

y = p(x).

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The Factor Theorem: If

€

(x − k) is a factor of a polynomial, then

€

x = k is a zero of the polynomial.

Conversely, if

€

x = k is a zero of a polynomial, then

€

(x − k) is a factor of the polynomial.

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Example 1: Find the zeros and x-intercepts of the graph of

€

p(x) =x

4−5x

2 + 4.

€

4−5x

2 + 4 = 0

2 − 4)(x

2 −1) = 0

(x + 2)(x − 2)(x +1)(x −1) = 0

x + 2 = 0 or x − 2 = 0 or x +1= 0 or x −1= 0

x = −2 or x = 2 or x = −1 or x =1

So the zeros are –2, 2, –1, and 1 and the x-intercepts are (–2,0), (2,0), (–1,0), and (1,0).

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The number of times a factor occurs in a polynomial is called the multiplicity of the factor. The

corresponding zero is said to have the same multiplicity. For example, if the factor

€

(x − 3) occurs to

the fifth power in a polynomial, then

€

(x − 3) is said to be a factor of multiplicity 5 and the

corresponding zero, x=3, is said to have multiplicity 5. A factor or zero with multiplicity two is

sometimes said to be a double factor or a double zero. Similarly, a factor or zero with multiplicity

three is sometimes said to be a triple factor or a triple zero.

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Example 2: Determine the equation, in factored form, of a polynomial

€

p(x) that has 5 as double

zero, –2 as a zero with multiplicity 1, and 0 as a zero with multiplicity 4.

€

p(x) = (x − 5)

2(x + 2)x

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Example 3: Give the zeros and their multiplicities for

€

p(x) = −12x

4 + 36x3 − 21x

€

−12x

4 + 36x3 − 21x

2 = 0

−3x

2(4x

2 −12x + 7) = 0

−3x

2 = 0 or 4x

2 −12x + 7 = 0

2 = 0 or x = −(−12)± (−12)

2−4(4)(7)

2(4)

x = 0 or x = 12± 144−112

8 = 12± 32

8 = 12±4 2

8 = 12

8 ± 4 2

8 = 3

2 ± 2

So 0 is a zero with multiplicity 2,

€

x = 3

2 − 2

2 is a zero with multiplicity 1, and

€

x = 3

2 + 2

2 is a zero

with multiplicity 1.

(Thomason - Fall 2008)

Because the graph of a polynomial is connected, if the polynomial is positive at one value of x and

negative at another value of x, then there must be a zero of the polynomial between those two values

of x.

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Example 4: Show that

€

p(x) = 2x3 − 5x

2 + 4 x − 7 must have a zero between

€

x =1 and

€

x = 2.

€

p(1) = 2(1)

3 − 5(1)

2 + 4(1) − 7 = 2(1) − 5(1) + 4 − 7 = 2 − 5 + 4 − 7 = −6

and

€

p(2) = 2(2)3 − 5(2)

2 + 4(2) − 7 = 2(8) − 5(2) + 8 − 7 =16 −10 + 8 − 7 = 7.

Because

€

p(1) is negative and

€

p(2) is positive and because the graph of

€

p(x) is connected,

€

p(x)

must equal 0 for a value of x between 1 and 2.

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If a factor of a polynomial occurs to an odd power, then the graph of the polynomial actually goes

across the x-axis at the corresponding x-intercept. An x-intercept of this type is sometimes called an

odd x-intercept. If a factor of a polynomial occurs to an even power, then the graph of the

polynomial "bounces" against the x-axis at the corresponding x-intercept, but not does not go across

the x-axis there. An x-intercept of this type is sometimes called an even x-intercept.

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Example 5: Use a graphing calculator or a computer program to graph

€

y = 0.01x

2(x + 2)3(x − 2)(x − 4)

4 .

–2 2 4

Because the factors

€

(x + 2) and

€

(x − 2) appear to odd

powers, the graph crosses the x-axis at

€

x = −2

and

€

x = 2.

Because the factors x and

€

(x − 4) appear to even

powers, the graph bounces against the x-axis at

€

x = 0

and

€

x = 4.

Note that if the factors of the polynomial were

multipled out, the leading term would be

€

0.01x10.

This accounts for the fact that both tails of the graph

go up; in other words, as

€

x → −∞,

€

Step-by-step explanation:

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Easy points. this is due tomorrow, the formula is given but im not sure how to solve. help? x

ozzi

$\sf{\qquad\qquad\huge\underline{{\sf Answer}}}$

Here we go ~

h = height of cone = 10 cm

r = radius of cone/sphere = ??

Volume of cone = 270 pi cm³

Volume of cone is :

$\qquad \sf \dashrightarrow \:v = 270 \pi$

$\qquad \sf \dashrightarrow \: \dfrac{1}{3} \cancel\pi {r}^{2} h = 270 \cancel\pi$

$\qquad \sf \dashrightarrow \:r {}^{2} \sdot10 = 270 \times 3$

$\qquad \sf \dashrightarrow \: {r}^{2} = 810 \div 10$

$\qquad \sf \dashrightarrow \: { {r}^{2} }^{} = 81$

$\qquad \sf \dashrightarrow \:r = \sqrt{81}$

$\qquad \sf \dashrightarrow \:r = 9 \: \: cm$

Now, let's calculate volume of solid sphere with same radius is ~

$\qquad \sf \dashrightarrow \:vol = \dfrac{4}{3} \pi {r}^{3}$

$\qquad \sf \dashrightarrow \:vol = \dfrac{4} {3} \sdot\pi \sdot {9}^{3}$

$\qquad \sf \dashrightarrow \:vol = \dfrac{4} {3} \sdot\pi \sdot 729$

$\qquad \sf \dashrightarrow \:vol = {4} {} \sdot243 \sdot\pi$

$\qquad \sf \dashrightarrow \:vol = 97 2\pi \: \: {cm}^{3}$

So, volume of the solid sphere in terms of pi is :

972 pi cm³

<u>note</u> : the solid figure attached below the cone is a hemisphere, so if the volume of hemisphere is asked then just dovide the result for sphere by 2. that is :

972pi / 2 = 486 pi cm³

4 0

2 years ago

A recent study of two vendors of desktop personal computers reported that out of 836 units sold by Brand A, 111 required repair,

iris [78.8K]

Answer:

Step-by-step explanation:

Hello!

The study variables are:

$X_A$ : The number of Brand A units sold that required repair.

$n_A= 836$

$x_A= 111$

$X_B$ : THe number of Brand B units sold that required repair.

$n_B= 739$

$x_B= 111$

1. Calculate the difference in the sample proportion for the two brands of computers, p^BrandA−p^BrandB =?.

The sample proportion of each sample is equal to the number of "success" observed xi divided by the sample size n:

^p $_A$ = $\frac{x_A}{n_A}= \frac{111}{836}= 0.1328$

^p $_B$ = $\frac{x_B}{n_B}= \frac{111}{739} =0.1502$

^p $_A$ - ^p $_B$ = 0.1328 - 0.1502= -0.0174

Note: proportions take numbers from 0 to 1, meaning they are always positive. But this time what you have to calculate is a difference between the two proportions so it is absolutely correct to reach a negative number it just means that one sample proportion is greater than the other.

2. What are the correct hypotheses for conducting a hypothesis test to determine whether the proportion of computers needing repairs is different for the two brands?

A. H0:pA−pB=0 , HA:pA−pB<0

B. H0:pA−pB=0 , HA:pA−pB>0

C. H0:pA−pB=0 , HA:pA−pB≠0

If you want to test whether the proportion of computers of both brands is different, you have to do a two-tailed test, the correct option is C.

3. Calculate the pooled estimate of the sample proportion, ^p= ?

To calculate the pooled sample proportion you have to use the following formula:

^p= $\frac{x_A+x_B}{n_A+n_B}= \frac{111+111}{836+739}= 0.14095 = 0.1410$

4. Is the success-failure condition met for this scenario?

A. Yes

B. No

The conditions that have to be met are:

$n_A\geq 30$ ⇒ Met

$n_A*p_A\geq 5$ ⇒ 836 * 0.1328= 111.4192; Met

$n_A*(1-p_A)\geq 5$ ⇒ 836 * (1 - 0.1328)= 727.5808; Met

$n_B\geq 30$ ⇒ Met

$n_B*p_B\geq 5$ ⇒ 739 * 0.1502= 110.9978; Met

$n_B*(1-p_B)\geq 5$ ⇒ 739 * (1-0.1502)= 628.0022; Met

All conditions are met.

5. Calculate the test statistic for this hypothesis test. ? =

$Z_{H_0}= \frac{(p'_A-p'_B)-(p_A-p_B)}{\sqrt{p'(1-p')[\frac{1}{n_A} +\frac{1}{n_B} ]} } = \frac{-0.0174-0}{\sqrt{0.1410*0.859*[\frac{1}{836} +\frac{1}{739} ]} }= -0.9902$

6. Calculate the p-value for this hypothesis test, p-value = .

This hypothesis test is two-tailed and so is the p-value, since it has two tails you have to calculate it as:

P(Z≤-0.9902) + P(Z≥0.9902)= P(Z≤-0.9902) + ( 1 - P(Z≤0.9902))= 0.161 + (1 - 0.839) = 0.322

7. What is your conclusion using α = 0.05?

A. Do not reject H0

B. Reject H0

The decision rule using th ep-value is:

If p-value > α, the decision is to not reject the null hypothesis.

If p-value ≤ α, the decision is to reject the null hypothesis.

The p-value= 0.322 is greater than α = 0.05, so the decision is to not reject the null hypothesis.

8. Compute a 95 % confidence interval for the difference p^BrandA−p^BrandB = ( , )

The formula to calculate the Confidence interval is a little different, because instead of the pooled sample proportion you have to use the sample proportion of each sample to calculate the standard deviation of the distribution:

( $p'_A-p'_B$ ) ± $Z_{1-\alpha /2}$ * $\sqrt{\frac{p'_A(1-p'_A)}{n_A} +\frac{p'_B(1-p'_B)}{n_B} }$