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marta [7]
3 years ago
15

Use the Remainder Theorem to completely factor p(x) = x ^3 + 7x ^2 + 11x + 5.

Mathematics
2 answers:
notka56 [123]3 years ago
5 0

Answer: (x+1)(x+1)(x+5)

Step-by-step explanation:

let f(x) = x^3 + 7x^2 + 11x +5

let x= -1

f(-1)= (-1)^3 + 7(-1)^2 + 11(-1) + 5

= -1 +7-11+5

=0

Therefore x + 1 is a factor

Then we proceed to divide

x^3 + 7x^2 + 11x +5 by x+1

_x^2+6x+5____

x+1√x^3 + 7x^2 + 11x + 5

-(x^3 + x^2)

____________________

6x^2 + 11x

- (6x^2 + 6x)

_______________________

5x +5

- (5x + 5)

_______________________

0

so after division,

x^2+6x+5 is also another factor, but we can further break it down into 2 factors.

let f(x) = x^2 + 6x + 5

let x= -1

f(-1)= (-1)^2 + 6(-1) + 5

= 1 - 6 + 5

= 0

Therefore (x+1) is also another factor,

we can use x+ 1 to divide

x^2 + 6x + 5 to get the next factor

_____x+5_____

x+1√x^2+6x +5

- (x^2+6x)

_____________

5x + 5

- (5x+5)

______________

0

Therefore x + 5 is another factors

(x+1)(x+1)(x+5) are the factors of the polynomial.

tigry1 [53]3 years ago
4 0

In order to use the remainder theorem, you need to have some idea what to divide by. The rational root theorem tells you rational roots will be from the list derived from the factors of the constant term, {±1, ±5}. When we compare coefficients of odd power terms to those of even power terms, we find their sums are equal, which means -1 is a root and (x +1) is a factor.

Dividing that from the cubic, we get a quotient of x² +6x +5 (and a remainder of zero). We recognize that 6 is the sum of the factors 1 and 5 of the constant term 5, so the factorization is

... = (x +1)(x +1)(x +5)

... = (x +1)²(x +5)

_____

The product of factors (x +a)(x +b) will be x² + (a+b)x + ab. That is, the factorization can be found by looking for factors of the constant term (ab) that add to give the coefficient of the linear term (a+b). The numbers found can be put directly into the binomial factors to make (x+a)(x+b).

When we have 1·5 = 5 and 1+5 = 6, we know the factorization of x²+6x+5 is (x+1)(x+5).

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A swimming pool is 20 feet wide and 40 feet long. If it is surrounded by a walkway of square tiles, each is 1 foot by 1 foot, ho
Lena [83]

Answer:

124 tiles

Step-by-step explanation:

2(20+40)

2(60)

120+4=124

5 0
2 years ago
Find the volume of the cubes or rectangular prisms in cubic centimeters.
DanielleElmas [232]

a) Volume of Rectangular prism is \frac{36}{125}\,\,cm^3

b) Volume of cube is \frac{8}{125}\,\,cm^3

c) Volume of cube is \frac{1}{125}\,\,cm^3

d) Volume of Rectangular prism is \frac{6}{125}\,\,cm^3

Step-by-step explanation:

Part a)

Volume of rectangular prism with

length= \frac{3}{5}\,\,cm

width= \frac{4}{5}\,\,cm

height = \frac{3}{5}\,\,cm

The formula used to find Volume of rectangular prism is:

Volume\,\,of\,\,rectangular\,\,prism=length*width*height

Putting values:

Volume\,\,of\,\,rectangular\,\,prism=\frac{3}{5} *\frac{4}{5} *\frac{3}{5}\\Volume\,\,of\,\,rectangular\,\,prism=\frac{36}{125}\,\,cm^3

So, Volume of Rectangular prism is \frac{36}{125}\,\,cm^3

Part b)

Volume of cube with side length of \frac{2}{5}\,\,cm

The formula used to find Volume of cube is:

Volume\,\,of\,\,cube=length^3

Putting value of length and finding volume:

Volume\,\,of\,\,cube=length^3\\Volume\,\,of\,\,cube=(\frac{2}{5})^3\\ Volume\,\,of\,\,cube=\frac{8}{125}\,\,cm^3

So, Volume of cube is \frac{8}{125}\,\,cm^3

Part c)

Volume of cube with side length of \frac{1}{5}\,\,cm

The formula used to find Volume of cube is:

Volume\,\,of\,\,cube=length^3

Putting value of length and finding volume:

Volume\,\,of\,\,cube=length^3\\Volume\,\,of\,\,cube=(\frac{1}{5})^3\\ Volume\,\,of\,\,cube=\frac{1}{125}\,\,cm^3

So, Volume of cube is \frac{1}{125}\,\,cm^3

Part d)

Volume of rectangular prism with

length= \frac{1}{5}\,\,cm

width= \frac{2}{5}\,\,cm

height = \frac{3}{5}\,\,cm

The formula used to find Volume of rectangular prism is:

Volume\,\,of\,\,rectangular\,\,prism=length*width*height

Putting values:

Volume\,\,of\,\,rectangular\,\,prism=\frac{1}{5} *\frac{2}{5} *\frac{3}{5}\\Volume\,\,of\,\,rectangular\,\,prism=\frac{6}{125}\,\,cm^3

So, Volume of Rectangular prism is \frac{6}{125}\,\,cm^3

Keywords: Volume

Learn more about Volume at:

  • brainly.com/question/6443737
  • brainly.com/question/12497249
  • brainly.com/question/12613605

#learnwithBrainly

6 0
3 years ago
Can you please answer both and show work
Effectus [21]

90*.1=9

90-9=81

81

190*.200=38

190-38=152

152

6 0
3 years ago
Read 2 more answers
How do you solve matrices
kiruha [24]

I've included two pictures showing steps on how to solve. Hopefully this helps more than words.

3 0
3 years ago
Suppose that a large mixing tank initially holds 500 gallons of water in which 50 pounds of salt have been dissolved. Another br
aliya0001 [1]

Answer:

A=1500-1450e^{-\dfrac{t}{250}}

Step-by-step explanation:

The large mixing tank initially holds 500 gallons of water in which 50 pounds of salt have been dissolved.

Volume = 500 gallons

Initial Amount of Salt, A(0)=50 pounds

Brine solution with concentration of 2 lb/gal is pumped into the tank at a rate of 3 gal/min

R_{in} =(concentration of salt in inflow)(input rate of brine)

=(2\frac{lbs}{gal})( 3\frac{gal}{min})\\R_{in}=6\frac{lbs}{min}

When the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min.

Concentration c(t) of the salt in the tank at time t

Concentration, C(t)=\dfrac{Amount}{Volume}=\dfrac{A(t)}{500}

R_{out}=(concentration of salt in outflow)(output rate of brine)

=(\frac{A(t)}{500})( 2\frac{gal}{min})\\R_{out}=\dfrac{A}{250}

Now, the rate of change of the amount of salt in the tank

\dfrac{dA}{dt}=R_{in}-R_{out}

\dfrac{dA}{dt}=6-\dfrac{A}{250}

We solve the resulting differential equation by separation of variables.  

\dfrac{dA}{dt}+\dfrac{A}{250}=6\\$The integrating factor: e^{\int \frac{1}{250}dt} =e^{\frac{t}{250}}\\$Multiplying by the integrating factor all through\\\dfrac{dA}{dt}e^{\frac{t}{250}}+\dfrac{A}{250}e^{\frac{t}{250}}=6e^{\frac{t}{250}}\\(Ae^{\frac{t}{250}})'=6e^{\frac{t}{250}}

Taking the integral of both sides

\int(Ae^{\frac{t}{250}})'=\int 6e^{\frac{t}{250}} dt\\Ae^{\frac{t}{250}}=6*250e^{\frac{t}{250}}+C, $(C a constant of integration)\\Ae^{\frac{t}{250}}=1500e^{\frac{t}{250}}+C\\$Divide all through by e^{\frac{t}{250}}\\A(t)=1500+Ce^{-\frac{t}{250}}

Recall that when t=0, A(t)=50 (our initial condition)

50=1500+Ce^{-\frac{0}{250}}50=1500+Ce^{0}\\C=-1450\\$Therefore the amount of salt in the tank at any time t is:\\A=1500-1450e^{-\dfrac{t}{250}}

4 0
3 years ago
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