Divide 16x3 – 12x2 + 20x – 3 by 4x + 5.

Answer:

[tex]4x^2 - 8x + 15 - \frac{78}{4x+5}[/tex]

Step-by-step explanation:

To solve polynomial long division problems like these, it's helpful to build a long division table. Getting used to building these can make problems like this much simpler to solve.

Begin by looking at the first term of the cubic polynomial.

What would we have to multiply [tex]4x + 5[/tex] by to get an expression containing [tex]16x^3[/tex]? The answer is [tex]4x^2[/tex], since [tex](4x + 5) \times 4x^2 = 16x^2 + 20x[/tex].

This is the first step of our long division, and we write out the start of our long division table like this:

[tex]{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,4x^2\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,16x^3 + 20x^2\\[/tex]

On the left is the divisor. On top is [tex]4x^2[/tex]. In the middle is the polynomial we are dividing, and on the bottom is the result of multiplying our divisor by

The next step is to subtract the bottom expression from the middle one, like so:

[tex]{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,4x^2\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\underline{16x^3 + 20x^2}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,0x^3 - 32x^2\\[/tex]

We are left with [tex]-32x^2[/tex]. The next thing to do is to add the next term of the polynomial we are dividing to the bottom line, like this:

[tex]{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,4x^2\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\underline{16x^3 + 20x^2}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 32x^2 + 20x\\[/tex]

Now we return to the beginning of the instructions, and repeat the process: namely, what would we have to multiply [tex]4x + 5[/tex] by to get an expression containing [tex]-32x^2[/tex]? The answer is [tex]-8x[/tex], and we fill out our long division table like so:

[tex]{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,4x^2 - \,\,\,\,8x\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\underline{16x^3 + 20x^2}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 32x^2 + 20x\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 32x^2 - 40x\\[/tex]

Once again, we subtract the bottom expression from the one above it, and include the next term of the divisor, like so:

[tex]{ }\qquad{ }\qquad{ }\quad{ }4x^2 - \,\,\,\,8x \,+ 15\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\underline{16x^3 + 20x^2}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 32x^2 + 20x\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{- 32x^2 - 40x}\\{ }\qquad{ }\qquad{ }\qquad{ }\qquad{ }\qquad{ }\,\,\,\,\,60x - 3\\[/tex]

And repeat. What do we multiply [tex]4x + 5[/tex] by to get an expression containing [tex]60x[/tex]? The answer is 15. Our completed long division table looks like this:[tex]{ }\qquad{ }\qquad{ }\quad{ }4x^2 - \,\,\,\,8x \,+15\\4x + 5\quad)\!\!\overline{\,\,\,16x^3 - 12x^2 + 20x - 3}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\underline{16x^3 + 20x^2}\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 32x^2 + 20x\\{ }\qquad{ }\quad{ }\quad{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{- 32x^2 - 40x}\\{ }\qquad{ }\qquad{ }\qquad{ }\qquad{ }\qquad{ }\,\,\,\,\,60x - 3\\{ }\qquad{ }\hspace{3cm}\,\,\underline{60x + 75}\\{ }\hspace{4.3cm}\,\,-78[/tex]

Now, the expression at the top,

[tex]4x^2 - 8x + 20x + 15[/tex]

is our quotient, and the last number, [tex]-78[/tex], is our remainder.

Hence we arrive at the solution of

[tex]\frac{16x^3-12x^2+20x-3}{4x+5} =4x^2 - 8x + 15 - \frac{78}{4x+5}.[/tex]

The diagram below is a scale drawing of a former's vegetable garden, where the length is 12 inches and the width is inches. The actual garden has a scale of 1 in 16 foot Select the following statements below the accurately represent the
cual garden.

What are the coordinates of the point on the directed line segment from (-8, -2) to
(6,2) that partitions the segment into a ratio of 1 to 3?

Will mark brainliest for correct answer, due soon!

2) f(x) = x2 - 6x +11
Degree:
Lead Coefficient:
End behavior:

a right circular cylinder is to be designed to hold 22 cubic inches of soft drink. Write the equation in r and h

How many nineths does it take to make the same as 1/3

5 Victor knows that 2 + 7 = 7+2 and 2•7= 7•2

He wonders if 2^7= 7^2

Fill in the blanks to explain whether 2^7= 7^2

To write 2^7as a repeated multiplication product, write

When 2^7is written as a repeated multiplication product, the only factor

To write 7^2 as a repeated multiplication product,

is

write

When 7^2 is written as a repeated multiplication product, the only factor

is

Check for common factors. Since 27 and 72 have

common

factor(s), 2

72

Complete the statement using <> or = -8/15 10/18

2.8125 rounded to the nearest tenth please!

A community college professor is interested in determining the relationship between the number of posts students make on Schoology and their respective grades. He planned to correlate the results from a sample from all his classes, however, due to time constraints, he decided to sample only from students with high grades. As his statistical consultant, state whether or not you support his plans and why.

in a lottery, players must match five numbers plus a bonus number. five white balls are chosen from 59 white balls numbered from 1 to 59, and one red ball (the bonus number) is chosen from 35 red balls numbered 1 to 5. how many different results are possible?

Which figure has the same area as the parallelogram?

I need help with this question

Find values of x and y
someone help me plz!!!!

Estimate the quotient of 1246 divided by 4

How many times smaller will a number become if is it decreased by 20%?

2. The time, t, for am ice cube to melt. is inversely proportional to the temperature, T, of the water the ice cube is placed. If it takes an ice cube 2 minutes to melt in 750F water, how long will it take an ice cube of the same size to melt in 800F water?

(WILL GIVE BRAINLIEST PLS HELP) Marco is saving for tuition costs at a state university. He deposited $8500 in a 4-year certificate of deposit that earns 7.25% per year. What equation represents this scenario?
f(x)=4*7.25^x
f(x)=8500*4^x
f(x)=1.0725*8500^x
f(x)=8500*1.0725^x

Pls help ASAP I’ll brainlest

3x - 2 y = 10 and 7 x + 2 y = 30

BRAINLIEST FOR WHOEVER GETS IT RIGHT

Simply the expression by combining like terms 6x+9+2x+4

PLEASE HELP ASAP, I ONLY HAVE AN HOUR!!!!!!!! HURRY!!!!!!!!!



If the sum of five consecutive odd integers is 405, what is the smallest of the five integers?

can anyone help plss

13 as prime or composite

Please answer correctly for heart and maybe brainiest
Here is a diagram of a person standing next to a tree.

The diagram shows two centimeter rulers.

The tree and the person are drawn to the same scale.

The person is approximately 1.65 m tall.

Using the scale diagram, estimate the height of the tree in meters

ASAP please i need help on this also ill give brainlist and please explain how you go the answer

What slope Line

pass through( 14, 4)( 21, 6 )

can someone give me the slope and y-intercept for this ?

Which algebraic expression matches the word phrase “A number minus seven is two”?
Group of answer choices

7 - x = 2

x - 7 = 2

x - 2 = 7

x = 7 - 2