Find the zeroes of the quadratic polynimial 3x²-x-4 and verify the relationship between the zeroes and the coefficien
Answers
Answer:
Step-by-step explanation:
3x^2-x-4=0
3x^2+3x-4x-4=0
3x(x+1)-4(x+1)=0
(3x-4)(x+1)=0
3x-4=0
x=4/3
x+1=0
x=-1
The sum of the zeroes of given polynomial=-b/a
-1+4/3=-(-1)/3
(-3+4)/3=1/3
1/3=1/3
Product of the zeroes of given polynomial=c/a
-1×4/3=-4/3
-4/3=-4/3
Answer :
Note:
★ The possible values of the variable for which the polynomial becomes zero are called its zeros .
★ A quadratic polynomial can have atmost two zeros .
★ The general form of a quadratic polynomial is given as ; ax² + bx + c .
★ If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then ;
• Sum of zeros , (α + ß) = -b/a
• Product of zeros , (αß) = c/a
★ If α and ß are the zeros of a quadratic polynomial , then that quadratic polynomial is given as : k•[ x² - (α + ß)x + αß ] , k ≠ 0.
★ The discriminant , D of the quadratic polynomial ax² + bx + c is given by ;
D = b² - 4ac
★ If D = 0 , then the zeros are real and equal .
★ If D > 0 , then the zeros are real and distinct .
★ If D < 0 , then the zeros are unreal (imaginary) .
Solution :
Here ,
The given quadratic polynomial is ;
3x² - x - 4 .
Clearly ,
a = 3
b = -1
c = -4
For finding zeros of the given quadratic polynomial , equate it to zero .
Thus ,
=> 3x² - x - 4 = 0
=> 3x² - 4x + 3x - 4 = 0
=> x(3x - 4) + (3x - 4) = 0
=> (3x - 4)(x + 1) = 0
=> x = 4/3 , -1
Now ,
• Sum of zeros = 4/3 + (-1)
= 4/3 - 1
= (4 - 3)/3
= 1/3
• -b/a = -(-1)/3 = 1/3
Clearly , Sum of zeros = -b/a
Also ,
• Product of zeros = (4/3)×(-1) = -4/3
• c/a = -4/3
Clearly , Product of zeros = c/a