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01.08.2022

How do i factor n2 - n - 56

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09.07.2023, solved by verified expert

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How do i factor n2 - n - 56, №18010554, 01.08.2022 09:51

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Mathematics

How do i factor n2 - n - 56

Step-by-step answer

P Answered by Specialist

${n}^{2} - n - 56 =$

(n - 8)(n + 7)

_____________________________________________

$a {x}^{2} + bx + c = 0$

$delta = {b}^{2} - 4ac$

$x(1) = \frac{ - b + \sqrt{delta} }{2a} \\$

$x(2) = \frac{ - b - \sqrt{delta} }{2a} \\$

_____________________________________________

${n}^{2} - n - 56 = 0$

$delta = ({ - 1})^{2} - 4(1)( - 56)$

delta = 1 + 224

delta = 225

Thus ;

$n(1) = \frac{ - ( -1) + \sqrt{225} }{2 \times 1} \\$

$n(1) = \frac{1 + 15}{2} \\$

$n(1) = \frac{16}{2} \\$

n(1) = 8

And

$n(2) = \frac{ - ( - 1) - \sqrt{225} }{2 \times 1} \\$

$n(2) = \frac{ 1 - 15}{2} \\$

$n(2) = - \frac{14}{2} \\$

n(2) = - 7

n(1) = 8

$n(1) - 8 = 0 \: \: \: (\alpha )$

$n - 8 = 0 \: \: \: \: ( \alpha )$ And

n(2) = - 7

$n(2) + 7 = 0 \: \: \: \: \: ( \beta )$

$n + 7 = 0 \: \: \: \: ( \beta )$

_____________________________________________

$\alpha \times \beta =$

(n - 8)(n + 7)

Mathematics

16. photon lighting company determines that the supply and demand functions for its most popular lamp are as follows: s(p) = 400 - 4p + 0.2p4 and d(p) = 2,800 - 0.0012p3, where p is the price. determine the price for which the supply equals the demand. a) $93.24 b) $100.24 c) $96.24 d) $99.24 17. write a polynomial function of minimum degree with real coefficients whose zeros include those listed. write the polynomial in standard form. 2, -4, and 1 + 3i (1 point) a) f(x) = x4 - 2x2 + 36x - 80 b) f(x) = x4 - 3x3 + 6x2 - 18x + 80 c) f(x) = x4 - 9x2

Step-by-step answer

P Answered by PhD

QUESTION 16

If the demand and supply are equal, then we equate the two functions in p and solve for p.

That is

S(p)=D(p)

$400 - 4p + + 0.00002 {p}^{4} = 2800 - 0.0012 {p}^{3}$

We can rearrange to obtain,

$0.00002 {x}^{4} + 0.0012 {x}^{3} - 4x - 2400 = 0$

$2 {p}^{2} + 1200 {p}^{3} - 400000p - 240000000 = 0$

The real roots of this polynomial equation are:

$p = - 118.26 \: p = 96.24$

Since price can not be negative, we discard the negative value ,

p = 96.24

The correct answer for question 16 is C.

QUESTION 17

We were given the solution to this polynomial as

x=2,x=-4, x=1+3i

We need to recognize the presence of the complex root and treat it nicely.

There is one property about complex roots of polynomial equations called the complex conjugate property. According to this property, if

a + bi

is a solution to

p(x)

then the complex conjugate

a - bi

is also a root.

Since

x = 1 + 3i

is a solution then,

x = 1 - 3i

is also a solution.

Therefore we have

f(x) = (x - 2)(x + 4)(x - (1+3i))(x - (1 - 3i)

$f(x) = ( {x}^{2} + 4x - 2x - 8)( {x}^{2} - (1 - 3i)x - (1 + 3i)x + (1 + 3i)(1 - 3i))$

We expand to obtain,

$( {x}^{2} + 4x - 2x - 8)( {x}^{2} - x + 3xi - x - 3xi + 1 + 9)$

Note that:

${i}^{2} = - 1$

$f(x) = ( {x}^{2} + 2x - 8)( {x}^{2} - 2x + 10)$

We now expand to obtain,

$f(x) = {x}^{4} - 2 {x}^{3} + 10 {x}^{2} + 2 {x}^{3} - 4 {x }^{2} + 20x - 8 {x}^{2} + 16x - 80$

We simplify further to obtain,

$f(x) = {x}^{4} + 2 {x}^{2} + 36x - 80$

The correct answer for question 17 is A.

QUESTION 18

-2i

is a zero of the polynomial,

f(x)=x^4-45x^2-196

then the complex conjugate

is also a zero,

This means that ,

(x+2i), (x-2i)

are factors of the polynomial.

The product of these two factors,

(x+2i), (x-2i) =x^2-(2i)^2=x^2+4

is also a factor , so we use it to divide and get the remaining factors.

see diagram for long division.

The above polynomial can therefore factored completely as,

f(x)=(x^2+4)(x^2-49)

Applying our knowledge from difference of two squares, we obtain,

f(x)=(x+2i)(x-2i))(x-7)(x+7)

Hence all the zeroes of these polynomial can be found by setting

(x+2i)(x-2i))(x-7)(x+7)=0

This implies,

x=-2i,x=2i,x=7,x=-7

The correct answer for question 18 is B

QUESTION 19

We were asked to find the horizontal and vertical asymptote of

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

To find the horizontal asymptote, divide the term with the highest degree in the numerator by the term with the highest degree in the denominators. That is the horizontal asymptote is given by,

$y=\frac{2x^2}{x^2}=2$

For vertical asymptote, equate the denominator to zero and solve for x.

x^2-1=

$\Rightarrow x=-1, x=1$

None of the options is correct, so the correct answer for question 19 is A.

QUESTION 20

We are converting,

$\frac{5\pi}{6}$

to degrees .

To convert from radians to degrees, multiply by,

$\frac{180\degree}{\pi}$

That is,

$\frac{5\pi}{6}=\frac{5\pi}{6} \times \frac{180\degree}{\pi}$

We simplify to obtain,

$\frac{5\pi}{6}=\frac{5}{1} \times \frac{180\degree}{1}=5\times30\degree =150\degree$

The correct answer is B.

QUESTION 21

Recall the mnemonics, SOH CAH TOA

The sine ratio is given by,

$\sin(B)=\frac{21}{75}=\frac{7}{25}$

From the diagram,

$\tan(B)=\frac{21}{72}=\frac{7}{24}$

The correct answer is C.

QUESTION 22

From the above diagram, We can determine the value of x using the sine or cosine ratio, depending on where the 17 is placed.

Using the cosine ratio, we obtain,

$\cos(58\degree)=\frac{17}{x}$

We can simply switch positions to make x the subject.

$x=\frac{17}{\cos(58\degree)}$

$x=\frac{17}{0.5299}$

x=32.08

Hence the correct answer is A.

QUESTION 23

Coterminal angles have the same terminal sides.

To find coterminal angles, we keep adding or subtracting 360 degrees.

See diagram.

$x=202\degree$

is coterminal with

$202\degree +360\degree =562\degree$

$202\degree -360\degree =-158\degree$

The correct answer is D.

See the attached file for continuation.

16. photon lighting company determines that the supply and demand functions for its most popular lam

Mathematics

Step-by-step answer

P Answered by PhD

QUESTION 16

If the demand and supply are equal, then we equate the two functions in p and solve for p.

That is

S(p)=D(p)

$400 - 4p + + 0.00002 {p}^{4} = 2800 - 0.0012 {p}^{3}$

We can rearrange to obtain,

$0.00002 {x}^{4} + 0.0012 {x}^{3} - 4x - 2400 = 0$

$2 {p}^{2} + 1200 {p}^{3} - 400000p - 240000000 = 0$

The real roots of this polynomial equation are:

$p = - 118.26 \: p = 96.24$

Since price can not be negative, we discard the negative value ,

p = 96.24

The correct answer for question 16 is C.

QUESTION 17

We were given the solution to this polynomial as

x=2,x=-4, x=1+3i

We need to recognize the presence of the complex root and treat it nicely.

There is one property about complex roots of polynomial equations called the complex conjugate property. According to this property, if

a + bi

is a solution to

p(x)

then the complex conjugate

a - bi

is also a root.

Since

x = 1 + 3i

is a solution then,

x = 1 - 3i

is also a solution.

Therefore we have

f(x) = (x - 2)(x + 4)(x - (1+3i))(x - (1 - 3i)

$f(x) = ( {x}^{2} + 4x - 2x - 8)( {x}^{2} - (1 - 3i)x - (1 + 3i)x + (1 + 3i)(1 - 3i))$

We expand to obtain,

$( {x}^{2} + 4x - 2x - 8)( {x}^{2} - x + 3xi - x - 3xi + 1 + 9)$

Note that:

${i}^{2} = - 1$

$f(x) = ( {x}^{2} + 2x - 8)( {x}^{2} - 2x + 10)$

We now expand to obtain,

$f(x) = {x}^{4} - 2 {x}^{3} + 10 {x}^{2} + 2 {x}^{3} - 4 {x }^{2} + 20x - 8 {x}^{2} + 16x - 80$

We simplify further to obtain,

$f(x) = {x}^{4} + 2 {x}^{2} + 36x - 80$

The correct answer for question 17 is A.

QUESTION 18

-2i

is a zero of the polynomial,

f(x)=x^4-45x^2-196

then the complex conjugate

is also a zero,

This means that ,

(x+2i), (x-2i)

are factors of the polynomial.

The product of these two factors,

(x+2i), (x-2i) =x^2-(2i)^2=x^2+4

is also a factor , so we use it to divide and get the remaining factors.

see diagram for long division.

The above polynomial can therefore factored completely as,

f(x)=(x^2+4)(x^2-49)

Applying our knowledge from difference of two squares, we obtain,

f(x)=(x+2i)(x-2i))(x-7)(x+7)

Hence all the zeroes of these polynomial can be found by setting

(x+2i)(x-2i))(x-7)(x+7)=0

This implies,

x=-2i,x=2i,x=7,x=-7

The correct answer for question 18 is B

QUESTION 19

We were asked to find the horizontal and vertical asymptote of

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

To find the horizontal asymptote, divide the term with the highest degree in the numerator by the term with the highest degree in the denominators. That is the horizontal asymptote is given by,

$y=\frac{2x^2}{x^2}=2$

For vertical asymptote, equate the denominator to zero and solve for x.

x^2-1=

$\Rightarrow x=-1, x=1$

None of the options is correct, so the correct answer for question 19 is A.

QUESTION 20

We are converting,

$\frac{5\pi}{6}$

to degrees .

To convert from radians to degrees, multiply by,

$\frac{180\degree}{\pi}$

That is,

$\frac{5\pi}{6}=\frac{5\pi}{6} \times \frac{180\degree}{\pi}$

We simplify to obtain,

$\frac{5\pi}{6}=\frac{5}{1} \times \frac{180\degree}{1}=5\times30\degree =150\degree$

The correct answer is B.

QUESTION 21

Recall the mnemonics, SOH CAH TOA

The sine ratio is given by,

$\sin(B)=\frac{21}{75}=\frac{7}{25}$

From the diagram,

$\tan(B)=\frac{21}{72}=\frac{7}{24}$

The correct answer is C.

QUESTION 22

From the above diagram, We can determine the value of x using the sine or cosine ratio, depending on where the 17 is placed.

Using the cosine ratio, we obtain,

$\cos(58\degree)=\frac{17}{x}$

We can simply switch positions to make x the subject.

$x=\frac{17}{\cos(58\degree)}$

$x=\frac{17}{0.5299}$

x=32.08

Hence the correct answer is A.

QUESTION 23

Coterminal angles have the same terminal sides.

To find coterminal angles, we keep adding or subtracting 360 degrees.

See diagram.

$x=202\degree$

is coterminal with

$202\degree +360\degree =562\degree$

$202\degree -360\degree =-158\degree$

The correct answer is D.

See the attached file for continuation.

Mathematics

The last time Robert filled up his car with gas, he paid $24.50 for 7 gallons. This time, he needs 15 gallons. If the price is the same, how much will he pay?

Step-by-step answer

P Answered by PhD

Cost of 7 gallons=$24.50

Cost of 1 gallon=24.50/7=3.5

Cost of 15 gallons=15*3.5=52.5

Cost of 15 gallons will be $52.5

Mathematics

An ice cream shop offers 5 different flavors of ice cream and 9 different toppings. How many choices are possible for a single serving of ice cream with one topping?

Step-by-step answer

P Answered by PhD

For 1 flavor there are 9 topping

Therefore, for 5 different flavors there will be 5*9 choices

No of choices= 5*9

=45

Mathematics

An online media service offers video and music downloads. Each download of a song costs $1. Each download of a video costs 5 times as much as the song download. You have a credit of $70. How many songs can you purchase after buying 12 videos?

Step-by-step answer

P Answered by PhD

The answer is in the image

Mathematics

The value of a collectible coin can be represented by the equation y = 2 x + 15, where x represents its age in years and y represents its total value in dollars. What is the value of the coin after 19 years? $2 $23 $38 $53

Step-by-step answer

P Answered by PhD

y=2x+15

where y=Value of coin

x=Age in years

Value of coin after 19 years=2*19+15

=$53

Therefore, Value after 19 years=$53

Mathematics

A 4300-N force from a car s engines produces an acceleration of 3.3 m/s2. What is the mass of the car? A. 4300 kg B. 1300 kg C. 14,000 kg D. 390 kg

Step-by-step answer

P Answered by PhD

F=ma

where F=force

m=mass

a=acceleration

Here,

F=4300

a=3.3m/s2

m=F/a

=4300/3.3

=1303.03kg

Mathematics

If a salesperson makes 6% commission, how much would they make from a $1,800 sale?

Step-by-step answer

P Answered by PhD

Salesperson will make 6% of 1800

=(6/100)*1800

=108

Salesperson will make $108 in $1800 sales

Mathematics

Mathematics, 26.03.2020 20:52, mohammadhosseinkarim David is building wooden picture frames. Each frame requires 3/4 feet of wood. If David started with 12 feet of wood and now has 6 feet of wood remaining, how many pictures frames has he built?

Step-by-step answer

P Answered by PhD

The wood before starting =12 feet

Left wood=6 feet

Wood used till now=12-6=6 feet

Picture frame built till now= 6/(3/4)

=8 pieces

Therefore, till now 8 pieces have been made.

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