Question

If x^3 + ax^2 – bx + 10 is divisible by x^3 -3x+ 2, find the values a and b.

Answer 1

Step-by-step explanation:

The Factor Theorem states that if a is the root of any polynomial p(x) that is if p(a)=0, then (x−a) is the factor of the polynomial p(x).

Let p(x)=x

3

+ax

2

−bx+10 and g(x)=x

2

−3x+2

Factorise g(x)=x

2

−3x+2:

x

2

−3x+2=x

2

−2x−x+2=x(x−2)−1(x−2)=(x−2)(x−1)

Therefore, g(x)=(x−2)(x−1)

It is given that p(x) is divisible by g(x), therefore, by factor theorem p(2)=0 and p(1)=0. Let us first find p(2) and p(1) as follows:

p(1)=1

3

+(a×1

2

)−(b×1)+10=1+(a×1)−b+10=a−b+11

p(2)=2

3

+(a×2

2

)−(b×2)+10=8+(a×4)−2b+10=4a−2b+18

Now equate p(2)=0 and p(1)=0 as shown below:

a−b+11=0

⇒a−b=−11.......(1)

4a−2b+18=0

⇒2(2a−b+9)=0

⇒2a−b+9=0

⇒2a−b=−9.......(2)

Now subtract equation 1 from equation 2:

(2a−a)+(−b+b)=(−9+11)

⇒a=2

Substitute a=2 in equation 1:

2−b=−11

⇒−b=−11−2

⇒−b=−13

⇒b=13

Hence, a=2 and b=13.

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