Question

If x³ + ax² - bx + 10 is divisible by x² - 3x

+2, find the values of a and b.


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Answer 1

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)=x² +ax² −bx+10 and g(x)=x² −3x+2

Factorise g(x)=x²−3x+2:

=x² −3x+2

=x²−2x−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)=1³ +(a×1² )−(b×1)+10=1+(a×1)−b+10=a−b+11

p(2)=2³ +(a×2² )−(b×2)+10

p(2)=8+(a×4)−2b+10

p(2)=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.

Answer 2

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• ⚘x3 + a x2 – bx + 10 is divisible by x2 – 3 x + 2
• ⚘Factors of x2 – 3 x + 2 are (x - 1) (x - 2)
• ⚘Hence, f(1) = 0
• ⚘1 + a - b + 10 = 0
• ⚘a - b = - 11..i)
• ⚘f(2) = 0
• ⚘23 + 22 a - 2b + 10 = 0
• ⚘8 + 4a - 2b = -10
• ⚘4a - 2b = - 18
• ⚘2a - b = -9..(ii)
• ⚘Subtracting (i) from (i)
• ⚘a - 2a - b + b = -11 + 9
• ⚘-a = -2
• ⚘a = 2
• Consider,
• ⚘a - b = -11
• ⚘2 - b = -11
• ⚘2 + 11 = b
• ⚘b = 13