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Answers
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.
The degree is even :
Product of roots = + ( coefficient of last term ) / ( coefficient of first term )
The degree is odd :
Product of roots = - ( coefficient of last term ) / ( coefficient of first term )
The polynomials of zeroes will be sloved by this:
