1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Veseljchak [2.6K]
2 years ago
5

Find all the zeros of the polynomial x^4+x^3-34x^2-4x+120 if two of its zeros are 2,-2

Mathematics
2 answers:
miskamm [114]2 years ago
7 0

Answer:

The zeroes are 2, -2, -6, 5.

Step-by-step explanation:

If 2 of the zeroes are 2 and -2 then (x - 2)(x + 2) is factor of the polynomial.

We can now do long division.

Note (x - 2)(x + 2) = x^2 - 4, so we have:

x^2 - 4 ( x^4 + x^3 - 34x^2 - 4x + 120 ( x^2 +   x  - 30 <--- Quotient

             x^4            - 4x^4

                      x^3 -  30x^2 - 4x

                      x^3                -4x

                                  -30x^2   + 120  

                                  -30x^2   + 120

                                           

x^2 +   x  - 30 = 0        

(x + 6)(x - 5) = 0

x = -6, 5.              

bezimeni [28]2 years ago
4 0

Answer:

5 and -6

Step-by-step explanation:

So when a polynomial has zeroes at a, then one of the factors (x-a). This is because in factored form, a polynomial can be written using it's factors as such: a(x\pm b)(x\pm c)(c \pm d)... and so on, with the number of factors depending on the degree. The sign is plus or minus because it can be either, it will depend on the sign of the zero. The reason the zeroes are factors, is because when one of the factors is zero it makes the entire thing zero because multiplying them out will always get 0, if you're multiplying by a zero. Anyways, if you have a factor, you can divide, by those factors. So let's do that.

So you have the two factors (x+2)(x-2). So let's divide by each factor. In this example I'll be using long division. So essentially what long division, is finding how many times the leading coefficient of the dividend (x^4+x^3-34x^2-4x+120) can be divided by the leading coefficient of the divisor ((x-2)(x+2)). And then whatever you get, that will be part of the quotient, then take whatever you got, and multiply it by the divisor and then subtract that product from the dividend and then continue this process until the dividend is equal to 0 or you can't divide any further./

Original equation:

(x+2)\div (x^4+x^3-34x^2-4x+120)

Divide the leading coefficients:

\frac{x^4}{x}=x^{4-1} = x^3

The x^3 will be part of the quotient, now multiply this by the divisor

x^3(x+2) = x^{3+1} + 2x^3 = x^4 + 2x^3

Subtract the product from the dividend

(x^4+x^3-34x^2-4x+120) - (x^4+2x^3) = x^4+x^3-34x^2-4x+120 - x^4 - 2x^3

Group like terms:

(x^4 - x^4) + (x^3 - 2x^3) -34x^2-4x+120

Simplify:

-x^3 -34x^2-4x+120

Divide leading coefficients:

\frac{-x^3}{x} = -x^{3-1} = -x^2

The -x^2 will be part of the quotient, now multiply this by the divisor:

-x^2(x+2) = -x^{2+1} -2x^2 = -x^3-2x^2

Subtract this from the dividend:

(-x^3 -34x^2-4x+120) - (-x^3 - 2x^2) = -x^3 -34x^2-4x+120 + x^3 + 2x^2

Group like terms:

(-x^3 + x^3) + (-34x^2 + 2x^2)-4x+120

Simplify:

-32x^2 -4x -120

Divide the leading coefficients:

-\frac{32x^2}{x}=-32x^{2-1} = -32x

This will be part of the quotient, now multiply this by the divisor:

-32x(x+2) = -32x^2 - 64x

Subtract this from the dividend

(-32x^2 -4x -120) - (-32x^2-64x) = -32x^2 -4x -120 + 32x^2 + 64x

Group like terms:

(-32x^2+32x^2) + (-4x+64x) -120

Simplify:

60x+120

Divide the leading coefficients:

\frac{60x}{x} = 60

This will be part of the quotient, multiply it by the divisor:

60(x+2) = 60x+ 120

Subtract this from the dividend:

(60x+120)-( 60x+120) = 0\\

Since this comes out perfectly to zero there is no remainder, and this is expected since it's a factor. You now take all the quotients from dividing the leading terms and you get x^3-x^2-32x+60. To keep the original equation you simply multiply it, and you get: (x+2)(x^3-x^2-32x+60). But what we really care about is the x^3-x^2-32x+60. We now divide it by the other factor x-2 using the same process

Original equation:

x^3-x^2-32x+60

Divide leading terms:

\frac{x^3}{x} = x^2

This will be part of the quotient, multiply it by the divisor

x^2(x-2) = x^3-2x^2

Subtract this from the dividend

(x^3-x^2-32x+60) - (x^3 - 2x^2) = x^2-32x+60

Dividing leading coefficients:

\frac{x^2}{x} = x

This will be part of the quotient, multiply it by the divisor:

x(x-2) = x^2-2x

Subtract this from the dividend:

(x^2-32x+60) - (x^2-2x) = -30x+60

Divide the leading terms:

\frac{-30x}{x} = -30

This will be part of the quotient, now multiply it by the divisor:

-30(x-2) = -30x+60

Subtract it from the dividend:

(-30x + 60) - (-30x + 60) = 0

Now take all the quotients of the leading terms and you get:

x^2+x-30

Now we can factor this quadratic to get the equation in completely factored form: We're looking for factors that multiply to -30 and add to 1. So we really need to look for factors that have a difference of 1, Or in other words |a-b| = 1. The sign can be determined after we fine the two factors. The factors 5 and 6 have a difference of 1, and since it needs to add to 1, the 5 has to be negative, and the 6 has to be positive.

So we get the factors: (x-5)(x+6). Setting each of them equal to zero gets you:

x-5 = 0

x = 5

x + 6 = 0

x = -6

So the other two zeroes are 5 and -6

You might be interested in
CAN SOMEONE HELP ME PLEASE
beks73 [17]

Answer:

x squared

Step-by-step explanation:

x exponent 4 can be written like this:

x*x*x*x

*=multiply

you can then divide it by x 3 times and... ye.

If you think my answer is good can you maybe give me brainliest? I am trying to achieve virtuoso rank.

4 0
2 years ago
Cuales son los terminos que siguen en cada sucesion:a)2,4,8,16,32,64,....B)1,3,5,7,9,....C)1,3,6,10,15,21,28,36,45
Elanso [62]

Answer:

a) Número siguiente = Número anterior * 2

b) Número siguiente = Número anterior + 2

c) Número siguiente = Número anterior + (n + 1)

Step-by-step explanation:

a) Para la secuencia 2,4,8,16,32,64, ....

El siguiente número es dos veces el número anterior.

Por ejemplo, 4 = 2 * 2, 8 = 4 * 2, 16 = 8 * 2 etc.

b) Para la secuencia 1,3,5,7,9, ....

El siguiente número es igual al número anterior + 2

Por ejemplo, 3 = 1 + 2, 5 = 3 + 2, 7 = 5 + 2 etc.

c) Para la secuencia1,3, 6,10,15,21,28,36,45

La diferencia entre los números aumenta en +1.

3-1 = 2

6-3 = 3

10-6 = 4

15-10 = 5

21-15 = 6

28-21 = 7

3 0
3 years ago
If a triangle is an equilateral triangle, then the triangle has exactly three 60°
ahrayia [7]

Step-by-step explanation:

In an equilateral triangle, all the three sides are equal as well as all the angles are equal. Let the angles be x.

We know that the sum of angles of a triangle is equal to 180 degrees. It means that,

x+x+x=180

3x=180

x=60°

Hence, if a triangle is an equilateral triangle, then the triangle has exactly three 60°  angles. Hence, △ABC is an equilateral triangle

3 0
3 years ago
Marcus purchases 75 shares of stock. A year after purchasing the stocks, the value of the stock is $2 less per share. Write a mu
katrin [286]
75(x-2) where x represents the amount originally paid

3 0
3 years ago
Given X(0, 2), Y(-2, 2), and Z(-2, 0), find the coordinates of X', Y', and Z' after a dilation with scale factor -4. ​
kotykmax [81]

Answer:

what is the origin of the dilation?

3 0
3 years ago
Other questions:
  • Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, three green ones, fo
    9·1 answer
  • Find the linear equation of a line that passes through the point (3,7) and is parallel to the line -8x 2y=16
    6·1 answer
  • A playground is being designed where children can interact with their friends in certain combinations. . If there is 1 child, th
    15·2 answers
  • What is the inverse of f(x)=x^2 - 2
    11·2 answers
  • A positive number multiplied or divided by a negative number is always<br> negative or positive
    5·2 answers
  • How many miles do you drive to school a statistical or a non-statistical question
    8·2 answers
  • Mmw-qdtk-wah come. ​
    11·2 answers
  • Question:
    14·1 answer
  • Can someone help me with this I need both area and perimeter
    10·2 answers
  • Katie just got her drivers license
    8·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!