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4 years ago

Find the remainder when f(x) = x3 − 14x2 51x − 22 is divided by x − 7.

Mathematics

2 answers:

Contact [7]4 years ago

5 0

The remainder is -8.

mel-nik [20]4 years ago

3 0

We can use at least two approaches in answering this question. But the easiest way is to use the $Remainder Theorem$

According to the Remainder Theorem, if
$(x - a)$
is not a factor of
$f(x)$
then evaluating the function at
$x = a$
gives us the Remainder when
$f(x)$
is divided by
$(x - a)$
That is
$f(a) = R$

For the given question, we have

$f(x) = {x}^{3} - 14 {x}^{2} + 51x - 22$
When this function is divided by
$x - 7$
The remainder is given by
$f(7) = R$
$f(7) = {7}^{3} - 14( {7})^{2} + 51(7) - 22$
$f(7) = 343 - 14( 49) + 51(7) - 22$
$f(7) = 343 - 686+ 357 - 22$
$f(7) = - 8$
$Hence the remainder is -8$

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So the easiest way to do this problem is to put your equation in linear form, meaning y = the rest of the equation.

I'm going to show the work to do that below:

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y = 2x - 4 (And just for neat purposes I'm going to put the 2x in front of the 4)

Now you have your equation ( y=2x-4 ) and you're ready to put it on the points of your graph.

Start on the point (0, -4) to represent the y axis.

y = 2x - 4

Then, your rise (2) over your run (1)

so your coordinates for a line will be:

(0, -4) (-2, 1) (1, 0)

You only need to plot these three to connect your line.

I appologize for my lack of visuals i understand this might be confusing, however if you have any further questions feel free to let me know!

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4 years ago

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3 years ago

What is the equation of the line that passes through (-3, -1) and has a slope of 3/5? Put your answer in slope-intercept form. y

zubka84 [21]

Answer:

Y=3/5x -4/5

Step-by-step explanation:

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3 years ago

. A fruit company recently released a new applesauce. By the end of its first year, profits on this product amounted to $24,700

lozanna [386]

Answer:

1) P(x) =14400x+10300 2)y=-4x+19 3) $y=\frac{-2x}{3} -14$ 4) x=-11

(Please check the pictures for better understanding)

Step-by-step explanation:

1) Firstly, let's organize the data. Let the dependent variable x be the years, and the Range P(x) the Profit. So arranging these linear functions y=ax+b and plugging the coordinates for each point (1, 24700) and (4, 67900). Goes:

24,700=x+b

67,900=4x +b

1.1)Solving for b, this Linear System of Equation by the Addition Method goes:

$24,700=x+b*(-4)\\67,900=4x+b\\----------\\-98,800=-4x-4b\\67,900=4x+b\\ -----------\\-30900=-3b\\\frac{3b}{3} =\frac{30900}{3}\\ b=10300$

1.2) Now, let's find the slope based on those two points (1, 24700) and (4, 67900) and using the formula to find the slope:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$m=\frac{67900-24700}{4-1}\\ m=\frac{43200}{3}\\ m=14400$

So now we can write this Linear Equation for the Profit:

P(x) =14400x+10300

2) An equation of the line perpendicular to 4y=x-8 has a slope equal to $m_{2}$ =- $\frac{-1}{m_{1}}$ the line. In this case, an equation perpendicular to 4y=x-8 .

Rearranging the given equation of the line 4y=x-8 goes:

$y=\frac{x}{4} -\frac{8}{4} \\ y=\frac{x}{4}-2$

2.1) The slope of this new line is equivalent to m=-4 since -4 is opposite inverse value for the slope of its perpendicular line (1/4).

If the line passes through the point (6,-5), then this point already belongs to this line.

Then, we can write.

-5=-4(6)+b

-5=-24+b

-5+24=-24+24+b

b=19

Now we can write the function:

y=-4x+19

3) Firstly, let's rearrange the equation. Into a General Equation of the Line y=ax+b

2x+3y=5

-2x+2x+3y=5-2x

3y=5-2x

$\frac{3y}{3} =\frac{5}{3} -\frac{2x}{3}$

$y=\frac{-2x}{3}+\frac{5}{3}$

3.2) Let's find out b, the linear coefficient by plugging the coordinates of the point (-3,-8).

Also, Parallel lines have the same value for the slope such as $m_{1}=m_{2}$

-8=-3(-2/3)+b

-8=2+b

-8-2=2-2+b

b=-10

Then the equation of this parallel line is:

$y=\frac{-2x}{3} -10$

4) Now on this last question, we have to find an equation of the line through the point (-11,-4) perpendicular to another line y=15

Suppose the slope of the first line is $m_{1}$ and the value of the slope of its perpendicular is $m_{2}=\frac{-1}{m_{1}}$

y=15 is a horizontal line with slope equals zero, we can rewrite it as y=0x+15.

Then the slope is zero for horizontal lines.

A perpendicular line to a horizontal line is a vertical one.

In addition to this, a vertical line passes through only one point in the x-axis. Then, there is no horizontal variation. This leads the formula of a slope to an undefinition for Real Set.

$m=\frac{y_{2}-y_{1}}{0}$

The only perpendicular line which passes through the point (-11,-4) to y=15 is the line x=-11.

7 0

3 years ago