Question 10 of 10
Use long division to find the quotient below.
(4x³+ 11x2 +9) + (x+3)
A. 4x²+x+3
OB. 4x²-2x+3
OC. 4x²-x+ 3
OD. 4x²+2x+3

Answer:

[tex]f(x)=x^3-12x^2+37x+130[/tex]

Step-by-step explanation:

So I'm assuming when you provided n=3, that means the degree is 3? So the first thing to know, is you can express a polynomial using it's factors as: [tex]f(x)=a(x\pm b)(x\pm b)(x\pm c)[/tex] where the sign of each factor depends on the sign of the factor... the point is it can be either. Notice the a? This usually will determine the stretch/compression of the polynomial, since sometimes the factors will have a coefficient for the x, but in this case I'm assuming all the coefficients of x are 1. So the next thing that is vital to know is that imaginary solutions come in conjugates. This means that if you have a zero at: [tex]a-bi \text{ then }a+bi \text{ is also a zero}[/tex]. So this means you have the 3 zeroes at x=-2, x=7+4i, x=7-4i. These are all the zeroes of the polynomial, since the Fundamental Theorem of Algebra states that a polynomial with degree n will have n solutions, which can be complex or real.

So the factor isn't going to be written as (x-2), it's going to be (x+2) since when you plug in -2 as x, it makes x+2 equal to 0.

The same thing applies for the two imaginary factors, so you're going to have the other two factors as (x-(7+4i)) and (x-(7-4i). You can instead think of it as ((x-7)+4i) and ((x-7)-4i) and you can use the difference of squares identity: [tex](a-b)(a+b)=a^2-b^2[/tex] where in this case a=x-7 and b=4i

So this gives you the equation: [tex]((x-7)-4i)((x-7)+4i) = (x-7)^2-(4i)^2[/tex]

Which becomes: [tex](x^2-14x+49)-16(-1) = x^2-14x+49+16 = x^2-14x+65[/tex]

So this gives us one of the factors: [tex]x^2-14x+65[/tex]

Now plug this in with the other factor and we get the equation:

[tex]f(x) = a(x+2)(x^2-14x+65)[/tex]

Now plug in 1 as x and make f(x) = 156, to solve for a

[tex]156=a(1+2)(1^2-14(1)+65)\\156=a(3)(1-14+65)\\156=a(3)(52)\\156=152a\\a=1[/tex]

So in this case, a=1, so that last step wasn't necessary, although I would check each time just in case a =/= 1.

Original equation

[tex]f(x) = (x+2)(x^2-14x+65)[/tex]

Multiply

[tex]f(x)=(x^3-14x^2+65x)+(2x^2-28x+130)\\[/tex]

Combine like terms:
[tex]f(x)=x^3+(-14x^2+2x^2)+(65x-28x)+130[/tex]

add like terms:

[tex]f(x)=x^3-12x^2+37x+130[/tex]

Answer:

[tex]f(x)=x^3-12x^2+37x+130[/tex]

Step-by-step explanation:

Complex Conjugate Theorem

For a polynomial p(x) with real coefficients, the complex zeros occur in conjugate pairs.  So if (a + bi) is a zero, then its conjugate (a - bi) is also a zero.

Given zeros:

-2 and (7 + 4i)

As one of the given zeros is a complex number, (7 - 4i) is also a zero.

Write each zero as part of a factor and multiply them together, adding a leading coefficient [tex]a[/tex]:

[tex]\begin{aligned}f(x) & = a(x+2)(x-(7+4i))(x-(7-4i))\\& = a(x+2)(x-7-4i)(x-7+4i)\\& = a(x+2)(x^2-7x+4ix-7x+49-28i-4ix+28i-16i^2)\\& = a(x+2)(x^2-14x+49-16i^2)\end{aligned}[/tex]

Remember that [tex]i^2=-1[/tex], therefore:

[tex]\begin{aligned}f(x) & = a(x+2)(x^2-14x+49-16(-1))\\& = a(x+2)(x^2-14x+49+16)\\& = a(x+2)(x^2-14x+65)\end{aligned}[/tex]

To find the value of the leading coefficient (a), use the given [tex]f(1)=156[/tex] :

[tex]\begin{aligned}f(1) & = 156\\\implies a(1+2)(1^2-14(1)+65) & = 156\\a(3)(52) & = 156\\156a & = 156\\\implies a & = 1\end{aligned}[/tex]

Therefore, the polynomial in factored form is:

[tex]f(x)=(x+2)(x^2-14x+65)[/tex]

Finally, expand the brackets:

[tex]\begin{aligned}f(x) & =(x+2)(x^2-14x+65)\\& = x^3-14x^2+65x+2x^2-28x+130\\& = x^3-14x^2+2x^2+65x-28x+130\\& = x^3-12x^2+37x+130\end{aligned}[/tex]

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Answer:

  (b)  (-3, 2, -5)

Step-by-step explanation:

The solution to the system of equations will satisfy every equation. If it fails to satisfy any equation, it is not a solution.

The system of equations does not seem to lend itself to simple solution by substitution or elimination. Hence, a reasonable strategy for finding the answer is to try the offered choices. Of course, a calculator can be used to find the answer almost as quickly.

Using the third equation, we can check the answer choices fairly easily. That equation involves the least number of arithmetic operations. Substituting for (x, y, z), we have ...

  a) 3(1) -(11) -(5) = -13 ≠ -6

  b) 3(-3) -(2) -(-5) = -6 . . . . . . a potential solution

  c) 3(1) -(8) -(0) = -5 ≠ -6

  d) 3(-1) -(3) -(4) = -10 ≠ -6

The only viable choice is (-3, 2, -5).

Check

In the other two equations, we have for this solution, ...

  2(-3) -2(2) +(-5) = -15 . . . . works in the first equation

  6(-3) -3(2) -(-5) = -18 . . . . works in the second equation

Answer:

[tex]\frac{1}{2}[/tex]

Step-by-step explanation:

plug 1 in for t

[tex]F(t)=4(\frac{1}{2^3^t})\\ F(1)=4(\frac{1}{2^3^*^1})\\ F(1)=4(\frac{1}{2^3})\\ F(1)=4(\frac{1}{8})\\ F(1)=\frac{4}{8} \\F(1)=\frac{1}{2}[/tex]

The measure of TV from the figure is 38 units

The given diagram in question is a kite. From the figure, the expression below is true;

6x - 3 = 39

Determine x

6x = 42

x= 7

TV = 2(2x+5)

TV = 2(2(7)+5)

TV = 2(19)

TV = 38 units

Hence the measure of TV from the figure is 38 units

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If graphed on the same grid, the diagrammatic representation of the parabola gives an upward curve.

The graph of the function is a diagrammatic representation of the slope, x-intercept, and y-intercept of the given function.

From the information given:

y = 2.5x²

Here,

x-intercept = (0,0)

y-intercept = (0,0)

Therefore, if graphed on the same grid, the diagrammatic representation of the parabola gives an upward curve.

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Answer:

-3 (v+4) + 6v

-3v - 12 + 6v

arranging like terms

-3v + 6v - 12

[tex]\Large\maltese\underline{\textsf{A. What is Asked}}[/tex]

-3(v+4)+6v, simplify

[tex]\Large\maltese\underline{\textsf{B. This problem has been solved!}}[/tex]

[tex]\multimap\bf{-3(v+4)+6v}=-3v-12+6v[/tex] | combine the like terms

[tex]\multimap\bf{3v-12}[/tex]

[tex]\cline{1-2}[/tex]

[tex]\bf{Result:}[/tex]

                [tex]\bf{=3v-12}[/tex]

[tex]\LARGE\boxed{\bf{aesthetic \not1\theta l}}[/tex]

Answer:

a and c = 115 and b and d = 65

Step-by-step explanation:

Hope this helps :)

Answer:

-76

Step-by-step explanation:

121 - y = 197

-121        -121

-y = 76

*-1   *-1

y = -76

The answer choice which must be true regarding the linear function is; The set must have a constant additive rate of change.

Since a linear function typically takes the slope-intercept form; f(x) = mx +c.

It therefore follows that the equation must have a constant slope m, which is the described additive constant rate of change.

It therefore follows that, the answer choice which is true regarding the linear function is therefore; The set must have a constant additive rate of change.

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The ordered pairs which is the solution of the system of Inequalities is;

x > 0 and y < 2

We are given the inequalities;

x + 2y > 4    ----(1)

3x - y < 2     -----(2)

Subtract 2y from both sides in eq 1 to get;

x > 4 - 2y   ----(3)

Put 4 - 2y for x in eq 2 to get;

3(4 - 2y) - y < 2

12 - 4y - y < 2

12 - 5y < 2

Subtract 12 from both sides to get;

-5y < -10

Divide both sides by -5 to get;

y < 2

Put 2 for y in eq 3 to get;

x > 4 - 2(2)

x > 0

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Since the grade of the numerator and the denominator is the same, then the limit exists and is distinct from 0. The limit of the expression is 4/7.

In this problem we must apply some algebraic handling and some known limits to determine whether the limit exists or not. The limit exists if and only if the result exists.

[tex]\lim_{x \to \infty} \frac{4\cdot x - 1}{7\cdot x + 3}[/tex]

[tex]\lim_{x \to \infty} \frac{4\cdot x - 1}{7\cdot x + 3} \cdot \frac{x}{x}[/tex]

[tex]\lim_{x \to \infty} \frac{4 - \frac{1}{x} }{7 + \frac{3}{x} }[/tex]

[tex]\lim_{x \to \infty} \frac{4}{7}[/tex]

4/7

Since the grade of the numerator and the denominator is the same, then the limit exists and is distinct from 0. The limit of the expression is 4/7.

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The sequences are classified, respectively, as:

Geometric, Arithmetic, Neither.

A geometric sequence is a sequence in which the result of the division of consecutive terms is always the same, called common ratio q.

In the first sequence, we have that:

[tex]q = \frac{45}{15} = \frac{15}{5} = \frac{5}{\frac{5}{3}} = \frac{\frac{5}{3}}{\frac{5}{9}} = 3[/tex]

Hence it is a geometric sequence.

In an arithmetic sequence, the difference between consecutive terms is always the same, called common difference d.

In the second sequence, we have that:

d = -4 - 1 = 1 - 6 = 6 - 11 = 11 - 16 = -5

Hence it is an arithmetic sequence.

[tex]\frac{4}{3} \neq \frac{3}{2}, 1 - \frac{1}{2} \neq 2 - 1[/tex]

Hence it is neither arithmetic nor geometric.

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Answer:

vii. x = 70°           viii. x = 60°

Step-by-step explanation:

vii.

The purple line drawn is parallel to the lines PQ and ST.

∴  ∠ a + 130° = 180°     [alternate angles]

⇒ ∠ a = 50°

x = a + 20°                  [Co-interior angles with angle [tex]x[/tex]]

⇒ x = 50° + 20°

⇒ x = 70°

viii.

The yellow line drawn and the line AB are parallel.

∴ n° + 130° = 180°        [Co-interior angles]

n = 50°

The green line and line DE are parallel.

∴ m° + 110° = 180°        [Co-interior angles]

m = 70°

n + m + x = 180°             [Angles on a straight line]

⇒ 50° + 70° + x = 180°

⇒ x = 60°

Answer:

vii) x = 70° viii) x = 60°

Step-by-step explanation:

Please refer to the attached photos for better understanding (Apologies for the terrible drawing.)

vii) Angle RSV + Angle RST  = 180° (Sum of angles in a straight line)

Angle RSV + 130°  = 180°

Angle RSV = 180° - 130°

                  = 50°

Angle RVS + Angle RSV + Angle SRV = 180° (Sum of angles in a triangle)

Angle RVS + 50°+ 20° = 180°

Angle RVS = 180° - 70° = 110°

Angle RVX + Angle RVS = 180° (Sum of angles in a straight line)

Angle RVX + 110° = 180°

Angle RVX = 180° - 110° = 70°

Angle x = Angle RVX = 70° (Corresponding Angles)

viii) Angle FDG + Angle CDE = 180° (Sum of angles in a straight line)

     Angle FDG + 110° = 180°

     Angle FDG = 180° - 110°

                         = 70°

Angle BGC = Angle FDG = 70° (Corresponding Angles)

Angle CBG + Angle ABC = 180° (Sum of angles in a straight line)

Angle CBG + 130° = 180°

Angle CBG = 180 - 130°

                   = 50°

Angle x + Angle CBG + Angle BGC = 180° (Sum of angles in a triangle)

Angle x + 50° + 70° = 180°

Angle x = 180° - 120° = 60°

Volume is a three-dimensional scalar quantity. The volume of concrete that will have to be poured into the pool deck is 6.0185 cubic yards.

A volume is a scalar number that expresses the amount of three-dimensional space enclosed by a closed surface.

Since an inch is equal to 1/12 of a foot. Therefore, the thickness of the pool deck is,

3 inches = 0.25 foot

If a pool deck of 650 square feet is to be laid with concrete 3" thick, then the volume of concrete that will be needed is,

Volume of concrete = 650 ft² × 0.25 ft = 162.5 ft³

Now, one yard is equal to 3 feet, therefore,

1 cubic foot = 1/27 cubic yards

162.5 cubic feet = 6.0185 cubic yards

Hence, the volume of concrete that will have to be poured into the pool deck is 6.0185 cubic yards.

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Answer:

32 seats per row

Step-by-step explanation:

Total number of seats in auditorium = 288

Number of rows in which seats evenly distributed = 9

Number of seats per row = 288/9= 32

The total number of possible fivey sequences are; 6

We are told we are dealing with a fivey sequence which means that all the terms of the sequence must be positive integers and they must add up to 5.

Now, the possible numbers that can be used are;

1, 2 and 3.

If we start with 3, number of possible sequences is; 1

If we start with1, number of possible sequences is; 3 ways

If we start with 2, number of possible sequences is; 2 ways

Thus;

Total number of fivey sequences is 1 + 2 + 3 = 6 possible sequences

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The number of tickets for adults is 210 and the number of tickets for children is 83.

It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.

If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.

Let x be the number of tickets for adults and y be the number of tickets for children.

From the problem:

x + y = 293

25.50x + 8y = 6019

After solving with the substitution method:

[tex]\rm 25.5\left(293-y\right)+8y=6019[/tex]

y = 83

x = 210

Thus, the number of tickets for adults is 210 and the number of tickets for children is 83.

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Answer:

330°

Step-by-step explanation:

Multiply by the conversion factor of [tex]\frac{180}{\pi }[/tex]

Answer:

180

Step-by-step explanation:

Let x be the first integer.

Let y be the 2nd integer.

Given info from the question,

x = 5% x y

x = 0.05y - equation 1

x + y = 63 - equation 2

Lets now substitute equation 1 into equation 2.

0.05y + y = 63

1.05y = 63

y = 63 / 1.05

= 60

Now we substitute y into equation 1.

x = 0.05(60)

= 3

Now since we know both x and y,

we can find the product of them.

xy = 60 * 3 = 180

the probability of rolling 4 or a number less than 6 is P = 5/6

The standard number cube has the outcomes {1, 2, 3, 4, 5, 6}

We want to find the probability of rolling 4 or a number less than 6. (4 is a number less than 6).

The outcomes that meet that condition are:

{1, 2, 3, 4, 5}

So 5 out of 6 outcomes meet the condition, then the probability is:

P(4 or less than 6) = 5/6

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Answer: -4

Step-by-step explanation:

[tex]\frac{1}{4}x-\frac{1}{8}=\frac{7}{8}+\frac{1}{2}x[/tex]

Multiplying both sides of the equation by 8,

[tex]2x-1=7+4x\\\\-1=7+2x\\\\-8=2x\\\\x=\boxed{-4}[/tex]

Answer:

23 years old

Step-by-step explanation:

Let x = age of younger sibling.

Then, the age of the older sibling is x + 9.

x + x + 9 = 55

2x = 46

x = 23

Answer: 23 years old

Step-by-step explanation:

•a reflection and a dilation

Answer:

sii esa misma vaina. loa a pero aya dur 7 dar s servicio rus ser rh r rhr ser ru r de dhd HD. dhs g PG15 BIT

 Answer:

 6x + 47

Step-by-step explanation:

Given following:

While solving these function sums, start from the right then head left.

⇒ f[g(x)]

⇒ f[x + 6]

⇒ 6(x + 6) + 11

⇒ 6x + 36 + 11

⇒ 6x + 47

Answer:

f(-2) = 3, f(0, -3) = -1

a) 32.4% probability that exactly 3 drivers text while driving if a police officer pulls over five drivers.

b)  2.76% probability the next driver texting while driving that the police officer pulls over is the fifth driver.

It is a branch of mathematics that deals with the occurrence of a random event.

Using Binomial distribution,

[tex]P(x) = C_{n , x} * p^{x}* (1-p)^ {n-x}[/tex]

We have, p= 0.52

A) probability that exactly 3 drivers text while driving if a police officer pulls over five drivers

[tex]P(x) = C_{5 , 3} * (0.52)^{3}* (0.48)^ {2}[/tex]

P(x) = 10* 0.140608* 0.2304

P(x)= 0.3239

32.4% probability that exactly 3 drivers text while driving if a police officer pulls over five drivers.

B) probability the next driver texting while driving that the police officer pulls over is the fifth driver

[tex]P(x) = C_{4 , 0} * (0.52)^{0}* (0.48)^ {4}[/tex]

P(x) = 1 * 1 * 0.053084

P(x) = 0.053084

So, 0.0531*0.52 = 0.0276

Hence, 2.76% probability the next driver texting while driving that the police officer pulls over is the fifth driver.

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The equation of the line through the points is y = 3/5 + 1/5.

First, we find the Slope using the formula: m = y2 - y1/x2 - x1

You may choose any two points but I will use these two for no reason at all.)

Points: (3,2) and (8,5)

m = y2 - y1/x2 - x1

m = 5 - 2/8 - 3

m = 3/5

Now that we have the slope, we can now find the Slope-Intercept Form by using the Slope Form Formula. (need at least one point and slope.)

Formula: y - y1 = m(x - x1)

Point: (3,2)

y - y1 = m(x - x1)

y - 2 = 3/5(x - 3)

y - 2 = 3/5x - 9/5

y - 2 = 3/5x - 9/5

y = 3/5x - 9/5 + 10/5 (2 can be is any fraction that consists of the numerator multiplied twice by the denominator, such as 2/1, 4/2, 8/4, etc.)

y = 3/5 + 1/5

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The graph which has the same end behaviour as the function given; f(x) = (10x³+x² +3)/(2x²+1) is; a linear graph whose slope is 5.

The function given can further be simplified by means of polynomial division in which case;

although not a perfect division; the quotient of the division is; (5x-0.5) and evidently, the function which would have the same end behaviour as the given function is a linear function whose slope is 5.

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Answer:

11

Step-by-step explanation:

since 30÷45 = 2/3 then

The space taken by a small book is equal to

the space taken by 2/3 a large book.

…………………………………………………

Calculating the number of large book corresponding to 23 small books :

= 23×(2÷3) + 3

= 18.333333333333 (large book)

Then ,Kevin can pack the shelf with (at most) : 30 - 19 = 11 more large books