Question

Find a and b so that the polynomial x3–10x2+ax+b is exactly
divisible by
the polynomials (x–1) and (x–2).

Answer 1

Required Answer:-

Question:

• Find a and b so that the polynomial x³ - 10x² + ax + b is exactly divisible by the polynomials (x - 1) and (x - 2)

Solution:

Let f(x) = x³ - 10x² + ax + b

Given that, (x - 1) is a factor of f(x). So, by factor theorem, f(1) = 0,

⟹ (1)³ - 10 × (1)² + a × (1) + b = 0

⟹ 1 - 10 + a + b = 0

⟹ a + b = 9 — (i)

Again, it's given that (x - 2) is a factor of f(x). So, by factor theorem, f(2) = 0,

⟹ (2)³ - 10 × (2)² + a × (2) + b = 0

⟹ 8 - 40 + 2a + b = 0

⟹ 2a + b = 32 —(ii)

Subtracting (i) from (ii), we get,

⟹ a = 32 - 9

⟹ a = 23

Putting the value of a in (i), we get,

⟹ 23 + b = 9

⟹ b = 9 - 23

⟹ b = -14

Hence,

1. a = 23
2. b = -14

So, the required polynomial is,

f(x) = x³ - 10x² + 23x - 14

Answer:

1. a = 23
2. b = -14

Learn More:

• Factor Theorem: If f(x) is a polynomial and α is a real number, then (x - α) is a factor of f(x) if f(α) = 0