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2 years ago

1. Which shows the factoring pattern for the sum of cubes?

Mathematics

1 answer:

Mazyrski [523]2 years ago

4 0

The solution to the questions are:

The expression that represents the factoring pattern is (a + b)(a² - ab + b²)
The solutions of the equation x³ + 2x² - 9x = 18 are x = -3, x = 3 and x = -2
The other roots are x = -3 and x = 2
The result of subtracting the expressions is - x³ + x² + 2x - 2
The largest multiplicity is 1

<h3>Factoring pattern for the sum of cubes</h3>

As a general rule, the sum of cubes x³ and y³ is:

x³ + y³= (x+y)(x² - xy + y²)

Replace x with a and y with b

a³ + b³= (a + b)(a² - ab + b²)

Hence, the expression that represents the factoring pattern is (a + b)(a² - ab + b²)

<h3>The solutions of the polynomial</h3>

The equation is given as:

x³ + 2x² - 9x = 18

Subtract 18 from both sides

x³ + 2x² - 9x - 18 = 0

Factorize the expression

x²(x + 2) - 9(x + 2) = 0

Factor out x + 2

(x²- 9)(x + 2) = 0

Express x²- 9 as a difference of two squares

(x + 3)(x - 3)(x + 2) = 0

Split

(x + 3) = 0 or (x - 3) = 0 or (x + 2) = 0

Remove brackets

x + 3 = 0 or x - 3 = 0 or x + 2 = 0

Solve for x

x = -3, x = 3 and x = -2

Hence, the solutions of the equation x³ + 2x² - 9x = 18 are x = -3, x = 3 and x = -2

<h3>The other roots of the polynomial</h3>

The polynomial equation is given as:

0 = x³ + 2x² - 5x - 6

Rewrite as:

x³ + 2x² - 5x - 6 = 0

Factorize the expression on the left-hand side

(x + 3)(x + 1)(x - 2) = 0

Split

x + 3 = 0 or x + 1 = 0 or x - 2 = 0

Solve for x

x = -3 or x = -1 or x = 2

Hence, the other roots are x = -3 and x = 2

<h3>Subtract the expressions</h3>

The expression is given as:

(x² - 1) - (x³ - 2x + 1)

Expand

x² - 1 - x³ + 2x - 1

Collect like terms

- x³ + x² + 2x - 1 - 1

Evaluate the like terms

- x³ + x² + 2x - 2

Hence, the result of subtracting the expressions is - x³ + x² + 2x - 2

<h3>The largest multiplicity of the polynomial</h3>

The polynomial is given as:

y = (x² - 1)(x + 2)(x + 4)

Express x² - 1 as a difference of two squares

y = (x - 1)(x + 1)(x + 2)(x + 4)

The power of each factor in the above expression is 1.

This means that the largest multiplicity is 1

You might be interested in

Plzz show workings while giving the answer

sukhopar [10]

Answer:

$\huge\boxed{\sf 4\ minutes\ 1.5\ seconds}$

Step-by-step explanation:

3 minutes 59.1 seconds = (3*60) + 59.1 seconds = 180 + 59.1 seconds

=> 239.1 seconds

$\rule[225]{225}{2}$

4 minutes 3.8 seconds = (4*60) + 3.8 seconds = 240 + 3.8 seconds

=> 243.8 seconds

$\rule[225]{225}{2}$

4 minutes 1.6 seconds = (4*60) + 1.6 seconds = 240 + 1.6 seconds

=> 241.6 seconds

$\rule[225]{225}{2}$

Mean = Sum of data / No. of Data

Mean = 239.1 + 243.8 + 241.6 / 3

Mean = 724.5 / 3

Mean = 241.5 seconds

<u>In Minutes and Seconds:</u>

= 241.5 / 60 = 4.025 minutes

= 4 minutes + 0.025 minutes

= 4 minutes + (0.025 * 60) seconds

= 4 minutes 1.5 seconds

$\rule[225]{225}{2}$

Hope this helped!

8 0

3 years ago

The Institute of Management Accountants (IMA) conducted a survey of senior finance professionals to gauge members’ thoughts on g

DochEvi [55]

Answer:

(1) The probability that the sample percentage indicating global warming is having a significant impact on the environment will be between 64% and 69% is 0.3674.

(2) The two population percentages that will contain the sample percentage with probability 90% are 0.57 and 0.73.

(3) The two population percentages that will contain the sample percentage with probability 95% are 0.55 and 0.75.

Step-by-step explanation:

Let <em>X</em> = number of senior professionals who thought that global warming is having a significant impact on the environment.

The random variable <em>X</em> follows a Binomial distribution with parameters <em>n</em> = 100 and <em>p</em> = 0.65.

But the sample selected is too large and the probability of success is close to 0.50.

So a Normal approximation to binomial can be applied to approximate the distribution of <em>p</em> if the following conditions are satisfied:

np ≥ 10
n(1 - p) ≥ 10

Check the conditions as follows:

$np= 100\times 0.65=65>10\\n(1-p)=100\times (1-0.65)=35>10$

Thus, a Normal approximation to binomial can be applied.

So, $\hat p\sim N(p, \frac{p(1-p)}{n})=N(0.65, 0.002275)$ .

(1)

Compute the value of $P(0.64$ as follows:

$P(0.64$

$=P(-0.20$

Thus, the probability that the sample percentage indicating global warming is having a significant impact on the environment will be between 64% and 69% is 0.3674.

(2)

Let $p_{1}$ and $p_{2}$ be the two population percentages that will contain the sample percentage with probability 90%.

That is,

$P(p_{1}$

Then,

$P(p_{1}$

$P(\frac{p_{1}-p}{\sqrt{\frac{p(1-p)}{n}}}$

$P(-z$

The value of <em>z</em> for P (Z < z) = 0.95 is

<em>z</em> = 1.65.

Compute the value of $p_{1}$ and $p_{2}$ as follows:

$-z=\frac{p_{1}-p}{\sqrt{\frac{p(1-p)}{n}}}\\-1.65=\frac{p_{1}-0.65}{\sqrt{\frac{0.65(1-0.65)}{100}}}\\p_{1}=0.65-(1.65\times 0.05)\\p_{1}=0.5675\\p_{1}\approx0.57$ $z=\frac{p_{2}-p}{\sqrt{\frac{p(1-p)}{n}}}\\1.65=\frac{p_{2}-0.65}{\sqrt{\frac{0.65(1-0.65)}{100}}}\\p_{2}=0.65+(1.65\times 0.05)\\p_{1}=0.7325\\p_{1}\approx0.73$

Thus, the two population percentages that will contain the sample percentage with probability 90% are 0.57 and 0.73.

(3)

Let $p_{1}$ and $p_{2}$ be the two population percentages that will contain the sample percentage with probability 95%.

That is,

$P(p_{1}$

Then,

$P(p_{1}$

$P(\frac{p_{1}-p}{\sqrt{\frac{p(1-p)}{n}}}$

$P(-z$

The value of <em>z</em> for P (Z < z) = 0.975 is

<em>z</em> = 1.96.

Compute the value of $p_{1}$ and $p_{2}$ as follows:

$-z=\frac{p_{1}-p}{\sqrt{\frac{p(1-p)}{n}}}\\-1.96=\frac{p_{1}-0.65}{\sqrt{\frac{0.65(1-0.65)}{100}}}\\p_{1}=0.65-(1.96\times 0.05)\\p_{1}=0.552\\p_{1}\approx0.55$ $z=\frac{p_{2}-p}{\sqrt{\frac{p(1-p)}{n}}}\\1.96=\frac{p_{2}-0.65}{\sqrt{\frac{0.65(1-0.65)}{100}}}\\p_{2}=0.65+(1.96\times 0.05)\\p_{1}=0.748\\p_{1}\approx0.75$