Question

If a and b are zeroes of the following quadratic polynomial, then find the values ​​of a + b and a,b


3x² - x - 4​

Answer 1

Answer:

Let us first factorize x

2

−4 as follows:

x

2

−4

=x

2

−2

2

=(x−2)(x+2)(∵a

2

−b

2

=(a−b)(a+b))

It is given that x

2

−4 is a factor of the polynomial f(x)=ax

4

+2x

3

−3x

2

+bx−4 that is (x−2)(x+2) are the factors of f(x)=ax

4

+2x

3

−3x

2

+bx−4 and therefore, x=−2 and x=2 are the zeroes of f(x).

Now, we substitute x=−2 and x=2 in f(x)=ax

4

+2x

3

−3x

2

+bx−4 as shown below:

f(−2)=a(−2)

4

+2(−2)

3

−3(−2)

2

+b(−2)−4

⇒0=16a−16−12−2b−4

⇒0=16a−2b−32

⇒2(8a−b−16)=0

⇒8a−b−16=0

⇒8a−b=16....(1)

f(2)=a(2)

4

+2(2)

3

−3(2)

2

+b(2)−4

⇒0=16a+16−12+2b−4

⇒0=16a+2b

⇒2(8a+b)=0

⇒8a+b=0....(2)

Adding equations 1 and 2:

(8a+8a)+(b−b)=16+0

⇒16a=16

⇒a=1

Substituting the value of a in equation 1, we get:

8a−b=16

⇒(8×1)−b=16

⇒8−b=16

⇒−b=16−8

⇒−b=8

⇒b=−8

Hence, a=1 and b=−8.

Step-by-step explanation:

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