Q.
Find the value of a and b so that x-1 and x+2 are factors of f(x)=2x^3+ax^2+bx-14
Answers
Step-by-step explanation:
x-1=0
x=1
f(1)=2(1)^3+a(1)^2+b(1)-14=0
= 2+a+b-14=0
= a+b-12=0
= a+b=12 _(1)
x+2=0
x=-2
f(-2)=2(-2)^3+a(-2)^2+b(-2)-14=0
= -16+4a-2b-14=0
= -30+4a-2b=0
=4a-2b=30
=2(2a-b)=30
=2a-b=15__(2)
on adding equation (1) and(2),
2a-b=15
a+b=12
b and -b will cancel each other
so,3a=27
a=27/3=9
a=9
put the value of a in equation (1),
a+b=12
9+b=12
b=12-9
b=3
hence, a=9 and b=3
Question:
Find the value of a and b so that x-1 and x+2 are factors of f(x)=2x³+ax²+bx-14.
Answer:
a = 9 , b = 3
Note:
• If (x-a) is a factor of the polynomial p(x) ,then x=a is a zero of polynomial p(x) and hence p(a) = 0.
• The possible values of x for which the polynomial p(x) becomes zero are called its zeros.
Solution:
The given polynomial is ;
f(x) = 2x³ + ax² + bx - 14
Also,
It is given that , (x-1) and (x+2) are the zeros of the given polynomial f(x) .
Since,
(x-1) is a factor of f(x) thus x = 1 is a zero of f(x).
Hence;
=> f(1) = 0
=> 2•1³ + a•1² + b•1 - 14 = 0
=> 2 + a + b - 14 = 0
=> a + b - 12 = 0
=> b = 12 - a -----(1)
Again,
Since , (x+2) is a factor of p(x) thus x = -2 is a zero of f(x) . Hence ;
=> f(-2) = 0
=> 2•(-2)³ + a•(-2)² + b•(-2) - 14 = 0
=> -16 + 4a - 2b - 14 = 0
=> 4a - 2b - 30 = 0
=> 2•(2a - b - 15) = 0
=> 2a - b - 15 = 0
=> b = 2a - 15 --------(2)
From eq-(2) and eq-(2) , we get ;
=> 12 - a = 2a - 15
=> 2a + a = 15 + 12
=> 3a = 27
=> a = 27/3
=> a = 9
Now,
Putting a = 9 in eq-(1) , we get ;
=> b = 12 - a
=> b = 12 - 9
=> b = 3
Hence,
The required values of a and b are 9 and 3 respectively .
Also refer to : https://brainly.in/question/16356316