Math, asked by Tomboyish44, 1 year ago

Obtain all the zeroes of the polynomial x⁴ + 6x³ + x² - 24x - 20 if two of its zeroes are 2 and -5

Grade 10, Polynomials.

Answers

Answered by brunoconti
18

Answer:

Step-by-step explanation:

Attachments:

Tomboyish44: Thank You so much!
Anonymous: (x+2) = 0; x = -2 not +2
Anonymous: btw Great answer :)
Answered by Anonymous
35

Answer:

The two other roots are - 2 or - 1 .

Step-by-step explanation:

Given polynomial is :

x⁴ + 6 x³ + x² - 24 x + 20  

Let f(x) = x⁴ + 6 x³ + x² - 24 x + 20  

Given that the roots are 2 and - 5 .

Let the other roots be a and b .

We know that the Sum of roots = - b / a

We also know that the Product of roots = e / a

Comparing x⁴ + 6x³ + x² - 24x - 20 with ax⁴ + bx³ + cx² + dx + e we get :

a = 1

b = 6

c = 1

d = - 24

e = - 20

According to the problem :

a + b + 2 + (-5) = - b / a

⇒ a + b - 3 = - 6 / 1

⇒ a + b - 3 = - 6

⇒ a + b = - 6 + 3

⇒ a + b = - 3  

⇒ a = - 3 - b ----------(1)

The product of roots is e / a

⇒ a × b × 2 × (-5) = - 20 / 1

⇒ - 10 ab = - 20

⇒ ab = (- 20 )/(- 10 )

⇒ ab = 2

Substituting the values gives :

( - 3 - b )( b ) = 2

⇒ - b² - 3 b - 2 =0

⇒ b² + 3 b + 2 = 0

⇒ b² + 2 b + b + 2 = 0

⇒ b ( b + 2 ) + 1 ( b + 2 ) = 0

⇒ ( b + 2 )( b + 1 ) = 0

⇒ either b = - 2

⇒ or b = - 1

The two other roots are - 2 and - 1 .

NOTE :

Zero product rule :

When ab = 0 either a or b = 0 .

This fact was used in the above problem :

( x + 2 )( x + 1 ) = 0

Either x + 2 = 0 , then x = - 2 .

Or x + 1 = 0 , then x = - 1 .

Another thing to note :

The degree of the polynomial

The degree of a polynomial is the highest power involved .

Here the degree is 4 since the highest power of x is 4 .

Note that the degree is important in determining the product of roots .

Degree is even :

Product of roots = + ( coefficient of last term ) / ( coefficient of first term )

Degree is odd :

Product of roots = - ( coefficient of last term ) / ( coefficient of first term )


Tomboyish44: Thanks a ton!
Anonymous: :)
Anonymous: Perfect answer :)
Anonymous: :)
sudhaddeviradiant: Thank you so much it is a perfect answer
Anonymous: Wlcm
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