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The polynomial functions for which (x+3) is a factor

A polynomial of degree n is a function of the form

f(x) = anx

n + an−1x

n−1 + . . . + a2x

2 + a1x + a0

where the a is a real number (sometimes called the coefficients of the polynomial). Although

this general formula might look quite complicated, particular examples are much simpler. For

example,

f(x) = 4x

3 − 3x

2 + 2

is a polynomial of degree 3, as 3 is the highest power of x in the formula. This is called a cubic

polynomial, or just cubic.

Theorem Polynomial Division: Suppose d(x) and p(x) are nonzero polynomials were

the degree of p is greater than or equal to the degree of d. There exist two unique polynomials,

q(x) and r(x), such that p(x) = d(x) q(x) + r(x), where either r(x) = 0 or the degree of r is

strictly less than the degree of d.

so, if we put x=-3 then f(x) must be equal to zero

here no function satisfies of none of the option is correct for [x+3] factor

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(x +3) is a factor of the function f(x)= x³ +5x²-3x³-10x²+2x-3.

Here, we are given 4 functions. If (x+3) is a factor of any of these equations, then x = -3 and y = 0 must be a solution. Let us check each function one by one-

1. f(x) = x²-12x³ +54x² - 108x+81

putting x = -3, we get-

f(-3) = -3²-12(-3)³ +54(-3)² - 108(-3)+81

= 9+324+486+324+81

= 1,224

Thus, (x +3) is not a factor.

2. f(x)=x²-3x³-x+3

putting x = -3, we get-

f(-3) = -3²-3(-3)³-(-3)+3

= 9+81+9+3

= 102

Thus, (x +3) is not a factor.

3. f(x)=x+2x¹-23x³-60x²

putting x = -3, we get-

f(-3) = -3+2(-3)¹-23(-3)³-60(-3)²

= -3-6+621+540

= 1,152

Thus, (x +3) is not a factor.

4. f(x)= x³ +5x²-3x³-10x²+2x-3

putting x = -3, we get-

f(-3) = -3³ +5(-3)²-3(-3)³-10(-3)²+2(-3)-3

= -27+45+81-90-6-3

= 0

Thus, (x +3) is a factor of the function f(x)= x³ +5x²-3x³-10x²+2x-3.

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Answer:

(0, 21)

Step-by-step explanation:

Stationary points occur when the gradient of a graph is zero.

Therefore, to find the x-coordinate(s) of the stationary points of a function, differentiate the function, set it to zero and solve for x.

Differentiate the given function:

[tex]\begin{aligned}y & = x^3+ax^2+b\\\implies \dfrac{\text{d}y}{\text{d}x}&=3 \cdot x^{3-1}+2 \cdot ax^{2-1}+0\\\implies \dfrac{\text{d}y}{\text{d}x}& = 3x^2+2ax\end{aligned}[/tex]

Set the differentiated function to zero:

[tex]\begin{aligned} \dfrac{\text{d}y}{\text{d}x}& =0\\ \implies 3x^2+2ax & = 0 \\ \implies x(3x+2a)&=0\end{aligned}[/tex]

Therefore the stationary points occur when:

[tex]\implies x=0[/tex]

[tex]\implies 3x+2a=0 \implies x=-\dfrac{2}{3}a[/tex]

There is a stationary point at (4, -11), therefore substitute x = 4 into the expression for x and solve for a:

[tex]\begin{aligned}x & = 4\\\implies -\dfrac{2}{3}a & = 4\\a & = -\dfrac{3}{2}(4)\\a & = -6\end{aligned}[/tex]

Substitute the found value of a and the point (4, -11) into the function and solve for b:

[tex]\begin{aligned}y & = x^3 + ax^2 + b\\\implies -11 & = (4)^3+(-6)(4)^2+b\\ -11 & = 64-96+b\\b & = -11-64+96\\ b & = 21\end{aligned}[/tex]

Therefore, the function is:

[tex]\boxed{y=x^3-6x^2+21}[/tex]

The other stationary point is when x = 0.  Therefore, to find the coordinates of this point, substitute x = 0 into the function:

[tex]\begin{aligned}y & =x^3-6x^2+21\\x=0\implies y & =(0)^3-6(0)^2+21\\ y & =21 \end{aligned}[/tex]

Therefore, the coordinates of the other stationary point are (0, 21).

To determine if a stationary point is minimum or maximum, differentiate the function again:

[tex]\begin{aligned}y & =x^3-6x^2+21\\\implies \dfrac{\text{d}y}{\text{d}x} & = 3x^2-12x\\\implies \dfrac{\text{d}^2y}{\text{d}x^2} & = 6x-12\\\end{aligned}[/tex]

Substitute the x-coordinate of the stationary point into the second derivative:

[tex]\begin{aligned} \dfrac{\text{d}^2y}{\text{d}x^2} & = 6x-12\\x=0 \implies \dfrac{\text{d}^2y}{\text{d}x^2} & = 6(0)-12= -12 \\\end{aligned}[/tex]

[tex]\textsf{As $\dfrac{\text{d}^2y}{\text{d}x^2} < 0$, stationary point $(0,21)$ is a maximum.}[/tex]

Differentiation Rules

[tex]\boxed{\begin{minipage}{4.8 cm}\underline{Differentiating $ax^n$}\\\\If $y=ax^n$, then $\dfrac{\text{d}y}{\text{d}x}=nax^{n-1}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{4cm}\underline{Differentiating a constant}\\\\If $y=a$, then $\dfrac{\text{d}y}{\text{d}x}=0$\\\end{minipage}}[/tex]

What if the median is zero?

A variable that can in principle be only zero or positive can only have mean zero if all values in practice are zero. On the other hand, such a variable can and will have median zero if more than half of the values are zero.

Can you have a median of one number?

Median of one number = the number itself.

The solution is (x, y, z) = (-1, 0, -2)

Given equations are:

2x−y+2z=−6 ---------------(1)

−3y+z=−2 ------------------(2)

2x−3z=4 -------------------(3)

From (2), z = -2+3y

Substitute this into (1) and (3)

(1)⇒ 2x-y+2(-2+3y) = -6

⇒ 2x+5y-4=-6

⇒2x+5y = -2

and

(3) ⇒ 2x-3(-2+3y)=4

⇒ 2x-9y+6 = 4

⇒ 2x-9y=-2

Now solve 2x+5y = -2 and 2x-9y=-2

Subtracting the equations above,

14y = 0

⇒y=0

Then from 2x-9y=-2, x = -1

From (2),z = -2+3y

⇒ z = -2

So the solution is (x, y, z) = (-1, 0, -2)

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Answer:

Hello,

Step-by-step explanation:

Only one solution: x=-2, y=3,z=-5

Answer:

consistent, independent; one

Step-by-step explanation:

The solution is (−2, 3, −5). Because the system has one solution, it is consistent and independent.

The expression f(x) = 3 · x² - 2 · x + 4 evaluated at x = a + h is equal to the expression f(x) = 3 · a² + 6 · a · h + 3 · h² - 2 · a - 2 · h + 4.

In this question we find a quadratic equation, which has to be evaluated at x = a + h and simplified afterwards. Then, the complete procedure is shown below:

f(x) = 3 · (a + h)² - 2 · (a + h) + 4

f(x) = 3 · (a² + 2 · a · h + h²) - 2 · a - 2 · h + 4

f(x) = 3 · a² + 6 · a · h + 3 · h² - 2 · a - 2 · h + 4

The expression f(x) = 3 · x² - 2 · x + 4 evaluated at x = a + h is equal to the expression f(x) = 3 · a² + 6 · a · h + 3 · h² - 2 · a - 2 · h + 4.

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Answer:

[tex]\overline{BC} \cong \overline{EF}[/tex]

Step-by-step explanation:

Corresponding parts of congruent triangles are congruent.

Answer:

Jojo is on a low carb diet

The independent variable in these equations based on how they are written is x, since y is a function of x here so y depends on the value of x, while x can be changed freely.

So, for each of these questions, plug in the given value of the independent variable for x.

Independent variable equal to 10: 0.1*10 + 5 = 6

Independent variable equal to 50: 0.1*50 + 5 = 10

Independent variable equal to 120: 0.1*120 + 5 = 17

The value of the given statement rounding to the nearest thousandth = 0.714.

Dependent occurrences are ones that are dependent on previous events. The outcomes of prior events have an impact on these occurrences. Dependent events are two or more occurrences that are dependent on one another. If one event is altered by chance, another is most likely to differ.

The independent and dependent factors are the two most crucial variables in a research experiment. The experimental has authority over the independent variable. The variable that varies in reaction toward the independent variable is known as the dependent variable.

P(A) = 0.74

P(B) = 0.85

P(A|B) = 0.84

Using  dependent events formula:

P(A∩B) = P(A/B) × P(B)

            = 0.84 X 0.85

            = 0.714

The value of the given statement rounding to the nearest thousandth = 0.714.

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Answer:

the answer is x=3

Step-by-step explanation:

Answer:

Please mark my answer as the brainliest, it would mean a lot to me

x = 5

Step-by-step explanation:

4x + 3 = 18 - x

3x + 3 = 18

3x = 15

x = 5

Answer:

160 mm.

Step-by-step explanation:

the average pen is 149mm. the average pen is 14.9cm. 160mm is closest.

It would have to be the 160mm.

This is because store-bought pens average 130 and 140mm/ 13 – 14 cm. This is closely followed by pens in the 140 – 150mm range.

Hope this helps!

Have a great day and God bless! :)

Answer:

Step-by-step explanation:

You want the values of three expenditures, given three relations between them.

(The first thing to realize is that the question isn't about averages at all. It is about values that are described as averages.)

Let f, j, c represent the (average) amounts spent on flowers, jewelry, and gift certificates, respectively. The three given relations are ...

  f + j + c = 53.42

  j = 4.40 + c

  f + c = 15.58 + j

Using the second equation to substitute for j in the third, we have ...

  f +c = 15.58 +(4.40 +c)

  f = 19.98 . . . . . . . . . . . . . . simplify

Using this and the second equation to substitute into the first, we have ...

  19.98 +(4.40 +c) +c = 53.42

  24.38 +2c = 53.42

  2c = 29.04

  c = 14.52

Then ...

  j = 4.40 +c = 4.40 +14.52 = 18.92

The average expenditures are ...

__

Additional comment

This solution is one of many ad hoc methods that could be used. There are also graphing and matrix methods available for solving a set of equations like this. The attachment shows a calculator matrix solution.

Answer:

Step-by-step explanation:

Given an assistant bricklayer laying 50 bricks an hour has worked 2 hours longer than a master bricklayer laying 80 bricks an hour, you want an equation stating they have laid the same number of bricks. You also want to know how long each has worked.

The output of each bricklayer is the product of the rate at which they lay bricks and the time for which they do it. If t is the time spent by the master bricklayer, their output will be 80t. The time spent by the assistant will be (t+2) hours, and their output will be 50(t+2).

Here is the equation that says their output is the same:

  80t = 50(t+2)

Solving the equation for t, we have ...

  80t = 50t +100

  30t = 100 . . . . . . . . . subtract 50t

  t = 3 1/3 . . . . . . . divide by 30

The master bricklayer has worked 3 1/3 hours when they have laid the same number of bricks.

The assistant bricklayer has worked 5 1/3 hours at that time.

__

Additional comment

Each will have laid (80)(3 1/3) = 266 2/3 bricks.

Reason:

7 units to the left means x becomes x-7

2 units up means y becomes y+2

Therefore, the old preimage (x,y) translates or shifts to the new image (x-7, y+2)

Example:

[tex](\text{x},\text{y}) = (3,5)\\\\(\text{x},\text{y})\to(\text{x}-7,\text{y}+2)\\\\(3,5)\to(3-7,5+2)\\\\(3,5)\to(-4,7)\\\\[/tex]

The preimage (3,5) moves to (-4,7) after shifting 7 units left and 2 units up.

The measurements are precise. Therefore, the correct option is D.

It should be noted that precision simply means how close a test is when it's repeated.

On the other hand, it should be noted that accuracy simply means how close the results are to the standard value.

Therefore, when the student does an experiment in a lab on the boiling point of water. In the 3 trials, the student gets the temperatures of 99.2, 99.5, 100.2 degrees, they're precise.

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W is the point which is non coplanar to plane k .

A group of points that are in the same plane are said to be coplanar. A coplanar line connects any two or three points. It's possible that four or more points are coplanar.

The coplanar points A, B, C, and D are displayed on the left side of the above figure. Additionally, there are other sets of coplanar points in the box to the right. For instance, the left side of the box's left side is the plane that contains the coplanar points P, Q, X, and W. There are four coplanar points on each of the box's six faces, however they are not the only coplanar point groups. Points Q, X, S, and Z, for instance, are coplanar despite the fact that the plane containing them isn't depicted.

Non-coplanar points are a collection of points that are not all located on the same plane.

Points P, Q, X, and Y in the previous figure are not coplanar. Q, X, and Y are located on the box's top, and P, Q, and X are located on the box's left side, however no flat surface has all four points.

since from the figure we can find out that except two points W , Y all the points are on the plane K .

so according to question and given figure we conclude that one of the point which in non coplanar to K is W.

points highlighted in the figure is non coplanar to point k .

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The simplified form of the function f(x) = (x² - 4x + 4) / (x² - 4) will be f(x) = (x - 2) / (x + 2).

A statement, principle, or policy that creates the link between two variables is known as a function. Functions are found all across mathematics and are required for the creation of complex relationships.

The function is given below.

f(x) = (x² - 4x + 4) / (x² - 4)

Simplify the equation, then we have

f(x) = (x² - 4x + 4) / (x² - 4)

f(x) = (x - 2)² / [(x - 2)(x + 2)]

f(x) = (x - 2) / (x + 2)

The simplified form of the function f(x) = (x² - 4x + 4) / (x² - 4) will be f(x) = (x - 2) / (x + 2).

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The average rate of change of the given function is 12.

The average rate of change of function f on interval [a,b] is:

f(b) -f(a)/b-a

we have interval [x₁,x₂]

[x₁,x₂] = [3,5]

f(3) = x²₊4x

f(3) = 3²+4(3)

f(3) = 21

f(5) = x²₊4x

f(5) = 5²+4(5)

f(5) = 45

f(x₂) -f(x₁)/x₂-x₁

⇒ 45-21/5-3

= 24/2

= 12

The average rate of change of function f(x) = x²+4x on interval [3,5]

is 12.

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Answer: an=3n-1    a₅=14   a₁₀₀=299

Step-by-step explanation:

a=2     d=3

[tex]\boxed {a_n=a_1+d(n-1)}\\\\a_n=2+3(n-1)\\\\a_n=2+3n-3\\\\a_n=3n-1\\\\a_{100}=3(100)-1\\\\a_{100}=300-1\\\\a_{100}=299\\\\a_5=3(5)-1\\\\a_5=15-1\\\\a_5=14[/tex]

Answer:

x = 4.8

Step-by-step explanation:

since the quadrilaterals are similar then the ratios of corresponding sides are in proportion , that is

[tex]\frac{MJ}{DA}[/tex] = [tex]\frac{LM}{CD}[/tex] ( substitute values )

[tex]\frac{x}{6}[/tex] = [tex]\frac{2.4}{3}[/tex] ( cross- multiply )

3x = 6 × 2.4 = 14.4 ( divide both sides by 3 )

x = 4.8

Answer:

Reflect over the x axis and than turn 90 degrees clockwise about the origin.

Answer:

[tex](f+g)(x)=8x^{2} +4x-4[/tex]

Step-by-step explanation:

[tex](f+g)(x)=3x^{2} +5x^{2} -2x+6x+4-8=8x^{2} +4x-4[/tex]

Hope this helps

Answer: 4/9

Step-by-step explanation:

[tex]\displaystyle\\(\frac{27}{8} )^{-\frac{2}{3}} =\\\\(\frac{3*3*3}{2*2*2})^{-\frac{2}{3}}=\\\\ (\frac{3^3}{2^3} )^{-\frac{2}{3}} =\\\\((\frac{3}{2})^3)^{-\frac{2}{3} } =\\\\(\frac{3}{2})^{-\frac{3*2}{3}} =\\\\(\frac{3}{2})^{-2}=\\\\(\frac{2}{3})^2 \\\\\frac{2^2}{3^2} =\\\\\frac{4}{9}[/tex]

Answer: -30 degrees fahrenheit

Step-by-step explanation: 18 + (-12) = -30

The table shows a proportional relationship between x and y.

In this question, we have been given a table which contains values of x and y.

We need to decide whether the table shows a proportional relationship between x and y.

x             y

17         2 5/6

5/6       5/36

6            1/6

19          3 1/6

First we convert the improper fraction in the y-column to proper fraction.

2 5/6 = 17/6

and 3 1/6 = 19/6

so, given table would be,

x             y

17          17/6

5/6       5/36

6            1/6

19          19/6

Consider the ratio x/y of each row:

x             y                x/y

17          17/6            1/6

5/6       5/36            1/6

6            1/6              1/6

19          19/6             1/6

As the ratio x/y is constant, x and y are directly proportional.

Therefore, the table shows a proportional relationship between x and y.

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Answer:

-8

Step-by-step explanation:

2x/7 =

2(12)/-3 =

24/-3 =

-8

Answer:

-8

Explanation:

Given expression:

[tex]\rightarrow \sf \dfrac{2x}{y}[/tex]

when  x = 12, y = -3

Substitute values inside expression:

[tex]\rightarrow \sf \dfrac{2(12)}{-3}[/tex]

multiply

[tex]\rightarrow \sf \dfrac{24}{-3}[/tex]

simplify

[tex]\rightarrow \sf -8[/tex]

Answer:

Nate can make 60 packages.

Step-by-step explanation:

7 1/2 is equivalent to 7 4/8, or 60/8

Divide 60/8 by 1/8 and you get 60.