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Tamiku [17]
3 years ago
7

What are all of the real roots of the following polynomial? f(x) x^4-24x^2-25

Mathematics
2 answers:
Delicious77 [7]3 years ago
4 0

Answer:

The real roots are  ±5

Step-by-step explanation:

It is given that,

f(x) = x^4 - 24x^2 - 25

<u>To find the real roots</u>

Let x^4 - 24x^2 - 25 = 0 ----(1)

Take y = x^2

Then eq (1) becomes,

y^2 - 24y - 25 = 0

By using splitting method we can write,

y^2 + y - 25y - 25 = 0

y(y + 1) - 25(y +1) = 0

(y + 1)(y - 25) = 0

(x^2 + 1 )(x^2 - 25) = 0

From (x^2 + 1 ) we get complex roots

x^2 = -1

x = √-1

x - 25 = 0 we get real roots

x = 25

x = ±5

Therefore the real roots are  ±5

Greeley [361]3 years ago
3 0

<u>Answer:</u>

x = ±5

<u>Step-by-step explanation:</u>

We are given the following polynomial function and we are to find all of its real roots:

x^4-24x^2-25

Let y=x^2 so we can now write it as:

y^2-24y-25

Factorizing it to get:

(y^2+y)+(-25y-25)

y\left(y+1\right)-25\left(y+1\right)

\left ( y + 1 \right ) \left ( y - 2 5 \right)

Substitute back y=x^2 to get:

\left ( x^2 + 1 \right ) \left ( x^2 - 25 \right )

\left ( x^2 + 1 \right ) \left ( x + 5 \right ) \left ( x - 5 \right )

The quadratic factor has only complex roots. Therefore, the real roots are x = ±5.

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Answer:

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Step-by-step explanation:

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3 years ago
Please help!
crimeas [40]

Answer:

The zeros are:

x =4, x=2, x = 5

  • The function has three distinct real zeros.

Hence, option (B) is true.

Step-by-step explanation:

Given the expression

h\left(x\right)=\left(x-4\right)^2\left(x^2-7x+\:10\right)

Let us determine the zeros of the function by putting h(x) = 0 and solving the expression

0=\left(x-4\right)^2\left(x^2-7x+10\right)

switch sides

\left(x-4\right)^2\left(x^2-7x+10\right)=0

as

x^2-7x+\:10=\left(x-2\right)\left(x-5\right)

so

\left(x-4\right)^2\left(x-2\right)\left(x-5\right)=0

Using the zero factor principle

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so

x-4=0\quad \mathrm{or}\quad \:x-2=0\quad \mathrm{or}\quad \:x-5=0

x =4, x=2, x = 5

Thus, the zeros are:

x =4, x=2, x = 5

It is clear that there are three zeros and all the zeros are distinct real numbers.

Therefore,

  • The function has three distinct real zeros.

Hence, option (B) is true.

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