X3+4x2-25x-100
find the polyomial
Answers
Answer :
The zeroes are -4 , -5 and 5.
Step-by-step explanation:
⇒ Cubic Polynomial :
- It is a polynomial of degree 3.
- General form :
ax³ + bx² + cx + d
- Relationship between zeroes and coefficients :
☆ Sum of zeroes = -b/a
☆ Sum of the product of zeroes taken two at a time = c/a
☆ Product of zeroes = -d/a
_______________________________
Given,
cubic polynomial, x³ + 4x² - 25x - 100
To find the zeroes of the polynomial, factorize it.
⇒ x³ + 4x² - 25x - 100 = 0
Find the common factor,
x² (x + 4) - 25 (x + 4) = 0
(x + 4) (x² - 25) = 0
(x + 4) (x² - 5²) = 0
we know, a² - b² = (a + b) (a - b)
(x + 4) (x + 5) (x - 5) = 0
The factors of the given polynomial are (x + 4) , (x + 5) and (x - 5)
=> x + 4 = 0 ; x = -4
=> x + 5 = 0 ; x = -5
=> x - 5 = 0 ; x = 5
The zeroes of the polynomial are -4 , -5 and 5
Verification :
- Put x = -4,
⇒ (-4)³ + 4(-4)² - 25(-4) - 100
⇒ -64 + 4(16) + 100 - 100
⇒ -64 + 64
⇒ 0
∴ -4 is a zero.
- Put x = -5,
⇒ (-5)³ + 4(-5)² - 25(-5) - 100
⇒ -125 + 4(25) + 125 - 100
⇒ -125 + 100 + 125 - 100
⇒ 0
∴ -5 is a zero
- Put x = 5,
⇒ 5³ + 4(5)² - 25(5) - 100
⇒ 125 + 4(25) - 125 - 100
⇒ 125 + 100 - 125 - 100
⇒ 0
∴ 5 is a zero
Hence verified!