The length of a pen is about

Using the laws of exponents, the equivalent expression would be 1 / 64.

To find the equivalent expression of ( 4 / 7 ) ⁻ ³  . 7 ⁻³, we can use the law of exponents.

Using this law, we get:

( 4 / 7 ) ⁻ ³  =  ( 1 / ( 4 / 7 )) ³

( 1 / ( 4 / 7 )) ³ = ( 7 / 4 ) ³

( 7 / 4) ³ x 7 ⁻³

( 7 / 4) ³ x ( 1 /7 ³ )

(( 7 ³ ) /( 4 ³ ) ) x ( 1/ 7 ³ )

( ( 7 ³ ) / ( 4 ³ ) )  x ( 1 / 7 ³ ) = 1 / (4 ³ )

1 / (4 ³ ) = 1 / 64

In conclusion, the equivalent expression is 1 / 64.

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The Binomial coefficients, we get:

(2x+5y)^3 = 8x^3 + 60x^2y + 150xy^2 + 125y^3

Therefore, (2x+5y)^3 expands to 8x^3 + 60x^2y + 150xy^2 + 125y^3.

To expand (2x+5y)^3, we can use the Binomial Theorem, which states that:

(a + b)^n = nC0a^n + nC1a^(n-1)b + nC2a^(n-2)b^2 + ... + nCn-1ab^(n-1) + nCn b^n

where nCk represents the binomial coefficient, which is the number of ways to choose k items from a set of n items.

In this case, we have:

a = 2x

b = 5y

n = 3

So, we can apply the Binomial Theorem as follows:

(2x+5y)^3 = 3C0 (2x)^3 + 3C1 (2x)^2(5y) + 3C2 (2x)(5y)^2 + 3C3 (5y)^3

Simplifying each term using the binomial coefficients, we get:

(2x+5y)^3 = 8x^3 + 60x^2y + 150xy^2 + 125y^3

Therefore, (2x+5y)^3 expands to 8x^3 + 60x^2y + 150xy^2 + 125y^3.

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The value of the missing side length x in the right triangle is 15 inches.

Pythagorean theorem states that the "square on the hypotenuse of a right-angled triangle is equal in area to the sum of the squares on the other two sides.

It is expressed as;

( hypotenuse )² = ( leg 1 )² + ( leg 2 )²

The image in the diagram is a right triangle:

Hypotenuse = 17 inches

Leg 1 = 8 inches

Leg 2 = x

To solve for x, we use the pythagorean theorem.

( hypotenuse )² = ( leg 1 )² + ( leg 2 )²

( leg 2 )² = ( hypotenuse )² - ( leg 1 )²

( leg 2 )² = ( 17 )² - ( 8 )²

( leg 2 )² = 289 - 64

( leg 2 )² = 225

Take the square roots

Leg 2 = √225

Leg 2 = 15 inches.

Therefore, the value of x is 15 inches.

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The L.C.M of c, 3c , 3 is 3c.

The L.C.M of c, 3c, and 3, we first need to factor each term:
c cannot be factored any further.
3c can be factored as 3 x c.
3 cannot be factored any further.
Next, we look for the highest common factors among the factors of these terms.

The only common factor is 3, which is included in both 3c and 3.
Therefore, the L.C.M of c, 3c, and 3 is 3c.
The L.C.M (Least Common Multiple) of c, 3c, and 3 can be found by analyzing the factors of each term.
For c, the only factor is c itself.
For 3c, the factors are 3 and c.
For 3, the only factor is 3.
Now, find the LCM by taking the highest power of each unique factor:
LCM(c, 3c, 3) = 3 * c = 3c

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1. The product of MG is as follows;

[tex]\left[\begin{array}{ccc}1&k_1+9&4k_2+36 \\2t_1&k_1t_1-17&3k_2-19t_1-11\\t_2+1&k_1-4t_2+5&k_2t_2-8\end{array}\right][/tex]

2. The values are k₁ = -9, k₂ = -9, t₁ = 0 and t₂ = -1 given that G = M⁻¹

Matrix M and matrix G are given as

M = [1, 4, 5; t₁, 3, -1; 1, t₂, 1] and

G = [2, k₁, -19; 1, -4, k₂; -1, 5, 11]

MG becomes

1×2 + 4×1 + 5×-1         1k₁+4×-4+ 5×5       1×-19+4k₂+5×11

t₁×2+3×1+-1×-1             t₁k₁+3×-4+-1×5       t₁×-19+3k₂+-1×11

1×2+t₂×1+1×-1               1k₁+t₂×-4+-1×5        1×-19+t₂k₂+1×11

=

1             k₁+9         4k₂+36

2t₁          k₁t₁-17        3k₂-19t₁-11

t₂+1        k₁-4t₂+5      k₂t₂-8

To solve G = M⁻¹ we know it is an identity matrix

I = | 1 0 0 |

    | 0 1 0 |

    | 0 0 1 |

We can equate the elements of MG and I to find the values of t₁, t₂, k₁, and k₂:

1 = 1,         k₁+9 = 0,       4k₂+36 = 0

2t₁ = 0     k₁t₁-17 = 1        3k₂-19t₁-11 = 0

t₂+1 = 0     k₁-4t₂+5 = 0   k₂t₂-8 = 1

k₁+9 = 0           4k₂+36 = 0            2t₁ = 0          t₂+1 = 0

k₁ = -9                 k₂ = -9                   t₁ = 0           t₂ = -1

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Answer:

-------------------

We know the tangent is perpendicular to radius.

Find the slope of the radius using its endpoints:

Perpendicular lines have negative reciprocal slopes, therefore the slope of the tangent is:

The matching choice is 2.

The distance travelled by the satellite in 36 hours would be: Distance travelled in 36 hours = (21991.5 / 24) × 36 = 549787.5 miles. Now, we can find the speed of the satellite as follows: Speed of the satellite = Distance / Time = 549787.5 / (36 × 3600) ≈ 4.76 miles/sec. Hence, the speed of the satellite is approximately 4.76 miles/sec (approximately 17136 miles/hr).

1. Find the distance from the satellite to the points of tangency Solution: Firstly, we need to draw a rough diagram of the scenario to easily understand it. Consider a satellite revolving around an alien planet in a synchronous orbit 18,000 miles above its surface.

We can see that B and D are the points of tangency of the satellite signal with the planet, and C is the center of the planet. Hence, the figure will look like the following: Image Source: Synchronous Orbit - Wikimedia Commons Now, we need to find the distance from the satellite to the points of tangency.

Hence, the distance from the satellite to the points of tangency, B and D, is 61861.9 miles (approximately 61862 miles).2. Find the percentage of the planet's equator within range of the satellite's signal Solution: We can see that the planet's equator can be represented by a circle with a radius of 3500 miles. Hence, the circumference of the circle would be 2π × 3500 miles ≈ 21991.5 miles.

We know that the satellite's signal can be received directly up to a distance of 61862 miles from it. Hence, the length of the portion of the equator within range of the satellite's signal would be twice the distance from the satellite to the point of tangency, which is 2 × 61862 ≈ 123724 miles. Now, we need to find the percentage of the planet's equator within range of the satellite's signal.

Hence, we need to find the time taken for the satellite signal to reach points B and E. We can see that the distance from the satellite to point B is 61862 miles and the distance from the satellite to point E is 3500 miles. Hence, the time taken for the satellite signal to reach point B would be: Time taken to reach point B = Distance / Speed = 61862 / 186000 ≈ 0.332 seconds Now, we need to find the time taken for the satellite signal to reach point E.

Hence, the time difference between the satellite signal reaching points B and E is approximately 0.07 seconds.4. Find the speed of the satellite Solution: We know that the satellite takes 36 hours to complete one revolution around the planet along with the point directly below it on the planet's surface. Hence, we need to find the distance travelled by the satellite in 36 hours to find its speed.

We can see that the distance travelled by the satellite in one revolution is equal to the circumference of the circle with a radius of 3500 miles, which is 2π × 3500 miles ≈ 21991.5 miles. Hence, the distance travelled by the satellite in 36 hours would be: Distance travelled in 36 hours = (21991.5 / 24) × 36 = 549787.5 miles.

Now, we can find the speed of the satellite as follows: Speed of the satellite = Distance / Time = 549787.5 / (36 × 3600) ≈ 4.76 miles/sec. Hence, the speed of the satellite is approximately 4.76 miles/sec (approximately 17136 miles/hr).

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the maximum area that can be enclosed with a budget of $6000 is 1,500,000 square meters.

Let's assume we purchase x meters of metal bars and y meters of wooden bars. The cost equation can be expressed as:

3x + 2y = 6000   (total cost equation)

We want to maximize the area, which is given by the equation:

Area = x * y

To solve this problem, we can use the method of substitution or elimination. Let's use substitution.

From the total cost equation, we can express x in terms of y:

x = (6000 - 2y) / 3

Now we can substitute this value of x into the area equation:

Area = [(6000 - 2y) / 3] * y

Simplifying further:

Area = (6000y - 2y^2) / 3

The x-coordinate of the vertex can be found using the formula:

x = -b / (2a)

In this case, a = -2 and b = 6000, so:

y = -6000 / (2 * -2) = 1500

Substituting this value of y back into the cost equation, we can find the corresponding value of x:

x = (6000 - 2 * 1500) / 3 = 1000

Therefore, with a budget of $6000, we can purchase 1000 meters of metal bars and 1500 meters of wooden bars, resulting in the maximum area.

Area = x * y = 1000 * 1500 = 1,500,000 square meters

So, the maximum area that can be enclosed with a budget of $6000 is 1,500,000 square meters.

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Answer:

Step-by-step explanation:

Given:

a₁ = 3

an+1 = (an)² + 1To find:

a₃Solution:

We can use the recursive formula given to find the value of a₃.a₂ = a₁ + 1² = 3 + 1 = 4

a₃ = a₂² + 1 = 4² + 1 = 17Therefore, the value of a₃ is 17.Answer: a₃ = 17

Answer:

See below for proof.

Step-by-step explanation:

[tex]\boxed{\textsf{Prove that}\;\;\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(240^{\circ}+ A) = \sin^3A}[/tex]

Rewrite 240° as (180° + 60°):

[tex]\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(180^{\circ}+60^{\circ}+ A)[/tex]

As sin(180° + x) = -sin(x), we can rewrite sin³(180° + 60° + A) as:

[tex]\sin^3(180^{\circ}+60^{\circ}+ A)=-\sin^3(60^{\circ}+ A)[/tex]

Substitute this into the expression:

[tex]\sin^3A + \sin^3(60^{\circ} + A) -\sin^3(60^{\circ}+ A)[/tex]

As the last two terms cancel each other, we have:

[tex]\sin^3A[/tex]

Hence proving that:

[tex]\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(240^{\circ}+ A) = \sin^3A[/tex]

As one calculation:

    [tex]\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(240^{\circ}+ A)[/tex]

[tex]=\sin^3A + \sin^3(60^{\circ} + A) + \sin^3(180^{\circ}+60^{\circ}+ A)[/tex]

[tex]=\sin^3A + \sin^3(60^{\circ} + A) -\sin^3(60^{\circ}+ A)[/tex]

[tex]=\sin^3A[/tex]

[tex]\hrulefill[/tex]

[tex]\boxed{\textsf{Prove that}\;\;\sin^3A + \sin^3(120^{\circ} + A) + \sin^3(240^{\circ}+ A) = -\dfrac{3}{4}\sin 3A}[/tex]

Use the sine and cos double angle identities to rewrite sin(3x) in terms of sin(x):

[tex]\begin{aligned}\sin(3x)&=\sin(2x+x)\\&=\sin2 (x)\cos (x)+\sin (x)\cos2 (x)\\&=(2\sin (x)\cos (x))\cos (x)+\sin (x)(1-2\sin^2 (x))\\&=2\sin (x)\cos^2 (x)+\sin (x)-2\sin^3 (x)\\&=2\sin (x)(1-\sin^2 (x))+\sin (x)-2\sin^3 (x)\\&=2\sin (x)-2\sin^3 (x)+\sin (x)-2\sin^3 (x)\\&=3\sin (x)-4\sin^3 (x)\end{aligned}[/tex]

Rearrange to isolate sin³x:

[tex]\begin{aligned}\sin(3x)&=3\sin (x)-4\sin^3 (x)\\\\4\sin^3 (x)&=3\sin (x)-\sin (3x)\\\\\sin^3 (x)&=\dfrac{3\sin (x)-\sin (3x)}{4}\end{aligned}[/tex]

Use this expression to rewrite the terms in sin³A on the left side of the equation:

   [tex]\sin^3A + \sin^3(120^{\circ} + A) + \sin^3(240^{\circ}+ A)[/tex]

[tex]=\dfrac{3\sin A-\sin3A}{4}+ \dfrac{3\sin (120^{\circ} + A)-\sin (3(120^{\circ} + A))}{4}+\dfrac{3\sin (240^{\circ}+ A)-\sin (3(240^{\circ}+ A))}{4}[/tex]

[tex]=\dfrac{3\sin A-\sin3A+3\sin (120^{\circ} + A)-\sin (360^{\circ} + 3A)+3\sin (240^{\circ}+ A)-\sin (720^{\circ}+ 3A)}{4}[/tex]

As sin(x ± 360°n) = sin(x), we can simplify:

[tex]\sin(360^{\circ}+3A) = \sin (3A)[/tex]

[tex]\sin(720^{\circ}+3A) = \sin (3A)[/tex]

Therefore:

[tex]=\dfrac{3\sin A-\sin3A+3\sin (120^{\circ} + A)-\sin (3A)+3\sin (240^{\circ}+ A)-\sin (3A)}{4}[/tex]

[tex]=\dfrac{3\sin A-3\sin3A+3\sin (120^{\circ} + A)+3\sin (240^{\circ}+ A)}{4}[/tex]

Factor out the 3 in the numerator:

[tex]=\dfrac{3\left(\sin A-\sin3A+\sin (120^{\circ} + A)+\sin (240^{\circ}+ A)\right)}{4}[/tex]

Rewrite 240° = 180° + 60°:

[tex]\sin(240^{\circ} + A) = \sin(180^{\circ} + 60^{\circ} + A)[/tex]

As sin(180° + x) = -sin(x), we can rewrite sin(180° + 60° + A) as:

[tex]- \sin(60^{\circ} + A)[/tex]

Therefore:

[tex]=\dfrac{3\left(\sin A-\sin3A+\sin (120^{\circ} + A)-\sin (60^{\circ}+ A)\right)}{4}[/tex]

As sin(120° + x) = sin(60° - x) then:

[tex]=\dfrac{3\left(\sin A-\sin3A+\sin (60^{\circ} -A)-\sin (60^{\circ}+ A)\right)}{4}[/tex]

As sin(60° - x) - sin(60° + x) = -sin(x), then:

[tex]=\dfrac{3\left(\sin A-\sin3A-\sin A\right)}{4}[/tex]

Simplify:

[tex]=\dfrac{-3\sin3A}{4}[/tex]

[tex]=-\dfrac{3}{4}\sin3A[/tex]

Hence proving that:

[tex]\sin^3A + \sin^3(120^{\circ} + A) + \sin^3(240^{\circ}+ A) = -\dfrac{3}{4}\sin 3A[/tex]

The amount of money that you would have by day 30 would be $ 2, 147 ,483 ,646.

To find the total amount of money you would have by day 30, we can use the formula for the sum of a geometric progression:

Sum = a x ( 1 - r ⁿ ) / ( 1 - r )

The sum would be the amount after 30 days which is:

Sum = 2 x  (1 - 2 ³⁰ ) / ( 1 - 2 )

Sum = 2 x ( 1 - 1, 073, 741,824 ) / ( - 1 )

Sum = 2 x (- 1 ,073 ,741,823) / ( -1 )

Sum = $ 2, 147 ,483 ,646

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Answer:

t=3.5

Step-by-step explanation:

Divide each term in 7t=24.5 by 7 and simplify.

The implications of Constructivist approach in teaching mathematics is that it helps the students to develop a better way to comprehend the content of what they are being taught.

A constructive approach of teaching mathematics believes that students construct knowledge rather than just passively take in information.

A constructive approach of teaching also believes that students integrate new knowledge with existing knowledge to create a deeper understanding of the mathematics being taught to them.

Therefore, constructive approach of teaching can help the students to comprehend the content of what they are being taught.

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Hello!

area

= (b x h)/2

= (18in x 7in)/2

= 126in²/2

= 63in²

Answer:

63 in²

Step-by-step explanation:

1/2 bh=area of the triangle

a. The rate of change of the relation is 2.

b. The value of n is 14.5.

In Mathematics and Geometry, the rate of change (slope) of any straight line can be determined by using this mathematical equation;

Rate of change (slope) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Rate of change (slope) = rise/run

Rate of change (slope) = (y₂ - y₁)/(x₂ - x₁)

By substituting the given data points into the formula for the rate of change (slope) of a line, we have the following;

Rate of change (slope) = (y₂ - y₁)/(x₂ - x₁)

Rate of change (slope) = (10 + 1)/(3.5 + 2)

Rate of change (slope) = 11/5.5

Rate of change (slope) = 2

Part b.

Next, we would determine the value of n as follows;

2 = (43 - 32)/(20 - n)

2(20 - n) = 11

40 - 2n = 11

2n = 29

n = 14.5

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Answer:

  $15,200

Step-by-step explanation:

You want to know the amount deposited at 1.5% that will earn the same simple interest in a year as $12,000 deposited at 1.9%.

The interest amount is given by the formula ...

  I = Prt

For the two accounts, the interest and the time are the same, so we have ...

  P1·r1 = P2·r2

  12000·1.9% = P2·1.5% . . . . . . . . . . . equate interest amounts for t=1

  P2 = 1.9/1.5·12000 = 15200 . . . . . . divide by 1.5%

Ruth deposited $15,200.

<95141404393>

The value of x is 5.

Using Angle Sum Property

The sum of the angles in a triangle is always 180 degrees. So, we can set up an equation:

9x-1 + 62 + 74 = 180

Simplifying the equation:

9x + 135 = 180

9x = 45

x = 5

Therefore, x is 5.

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The seagull is approximately 47.86 feet above sea level, and the man's binoculars are approximately 46.25 feet away from the dolphin.

Let's denote the height of the seagull above sea level as "h" and the distance between the man's binoculars and the dolphin as "d."

For the seagull:

We have the angle of elevation as 27 degrees and the distance from the man's binoculars as 94 feet.

Using the tangent function, we can write:

tan(27 degrees) = h / 94

Solving for h, we have:

h = 94 × tan(27 degrees)

For the dolphin:

We have the angle of depression as 46 degrees and the distance from the man's binoculars as 44 feet (lenses level).

Using the tangent function, we can write:

tan(46 degrees) = h / d

Solving for d, we have:

d = h / tan(46 degrees)

Now, let's calculate the values:

h = 94 × tan(27 degrees) ≈ 47.86 feet

d = (47.86 feet) / tan(46 degrees) ≈ 46.25 feet

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The equation that represents the situation is a = b- 4.

Since the plant grows at a constant rate, we can assume that the height of the plant is increasing linearly with time.

Let's use the data from Week 1 and Week 2 to find the rate of growth for each plant:

For Plant A: Growth rate = (24 cm - 22 cm) / (2 weeks - 1 week) = 2 cm/week

For Plant B: Growth rate = (28 cm - 26 cm) / (2 weeks - 1 week) = 2 cm/week

Since the growth rate is constant, we can use the equation of a line to model the height of each plant over time:

For Plant A: a = 2t + b, where t is the time in weeks and b is the initial height of the plant.

For Plant B: b = 2t + c, where c is the initial height of Plant B.

We can find the values of b and c by substituting the data from Week 1 into these equations:

For Plant A: 22 = 2(1) + b, so b = 20.

For Plant B: 26 = 2(1) + c, so c = 24.

Now we can substitute these values into the equations for Plant A and Plant B:

Plant A: a = 2t + 20

Plant B: b = 2t + 24

To find an equation that gives Plant A's height in terms of Plant B's height, we can solve the equation for t in terms of b:

b = 2t + 24

2t = b - 24

t = (b - 24) / 2

Then we can substitute this expression for t into the equation for Plant A:

a = 2t + 20

a = 2[(b - 24) / 2] + 20

a = b - 24 + 20

a = b - 4

So the equation we were looking for is:

a = b - 4

Therefore, to find Plant A's height in centimeters given Plant B's height in centimeters, we simply subtract 4 from Plant B's height.

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The equation of h(x) in vertex form is given as follows:

h(x) = (x + 2)² - 3.

The quadratic function of vertex(h,k) is given by the rule presented as follows:

y = a(x - h)² + k

In which:

The vertex is the turning point of the function, where it changes from decreasing to increasing, or vice versa, hence it's coordinates are given as follows:

(-2,-3).

Then the equation is:

h(x) = a(x + 2)² - 3.

When x = 0, h(x) = 1, hence the leading coefficient a is obtained as follows:

1 = a(2)² - 3

4a = 4

a = 1.

Hence the equation is:

h(x) = (x + 2)² - 3.

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After considering the given data we conclude that the value generated after performing divison of the given expression is 18x, under the condition that we follow the principles of divison.

Let us proceed by evaluating the given expression by applying division
4x(3+6) divided by 2
= 4x9/2
= 36x/2
= 18x
Division is considered one of the four basic mathematical operations, then the other three are addition, subtraction, and multiplication.
In short , division is referring to the breaking down of a given bigger number or expression into simpler yet smaller forms to meet the criteria of performing a function.
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Answer:

24

Step-by-step explanation:

As x is now equal to 5, all you have to do is substitute it into were x is in the f(x) formula given.

x^2-1 turns into (5)^2-1=

25-1=

24

The difference between the interquartile range of Matt's remaining vacation days an Linda's remaining vacation days is given as follows:

-3.5.

The interquartile range of a data-set is given by the difference of the third quartile by the first quartile of the data-set.

Matt's ordered data-set is given as follows:

5, 9, 11, 12.

Hence the quartiles are:

Hence the IQR is of:

IQR = 11.5 - 7 = 4.5.

Linda's ordered data-set is given as follows:

0, 6, 9, 13.

The quartiles aret

The IQR is of:

IQR = 11 - 3 = 8.

Hence the difference is of:

4.5 - 8 = -3.5.

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Answer:

25/51

Step-by-step explanation:

If the first card was a heart, we've drawn one red card already.

The ratio of black to red cards in the deck is now 26:25.

On the second draw, there will be only 51 cards to choose from, rather than 52 because you have already taken out a card. Therefore, you only have a 25/51 chance of drawing a red card compared to the 26/52 or 1/2 chance on the first draw.

Answer is 25/51.

Factor of x ^2+7x−18 is (x−2)(x+9).

We have,

In mathematics, factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.

We are given a polynomial expression in terms of variable x as:

x² - 7x + 18

so, we have,

x ^2+7x−18

=x ^2−2x+9x−18

=x(x−2)+9(x−2)

=(x−2)(x+9)

∴ x ^2+7x−18=(x−2)(x+9)

Hence, Factor of x ^2+7x−18 is (x−2)(x+9)

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complete question:

Factor completely x² - 7x + 18.

Prime

(x-9)(x-2)

(x-9)(x+2)

(x + 9)(x+2)

The equation of the polynomial is f ( x ) = -13.5 ( x + 2 )²( x - 3 )

Given data ,

The equation of degree 3 polynomial function with real coefficients

And ,  zeros x = - 2 with multiplicity 2 and x = 3 with multiplicity 1

where function passes through the point (1, 54)

If a polynomial function has a zero x = a with multiplicity k, then the factor (x - a)^k appears in its factored form.

Therefore, a degree 3 polynomial function with zeros x = -2 with multiplicity 2 and x = 3 with multiplicity 1 can be written in factored form as:

f(x) = a(x + 2)²(x - 3)

where a is a constant factor. To find the value of a, we use the fact that the function passes through the point (1, 54):

f(1) = a(1 + 2)²(1 - 3) = 54

a(-1)²(-2) = 54

-4a = 54

Divide by -4 on both sides , we get

a = -13.5

Hence , the equation of the degree 3 polynomial function with the given zeros and passing through the point (1, 54) is f ( x ) = -13.5 ( x + 2 )²( x - 3 )

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