Question

for what value of k the expression x3+x4-14x-k is exactly divisible by x+2​

Answer 1

Given:-

• p(x) = x⁴ + x³ - 14x - k
• x + 2 is a factor of p(x)

To find:-

• The value of k.

Answer:-

If p(x) = x⁴ + x³ - 14x - k is divisible by (x + 2), that means (x + 2) is a factor of p(x).

From factor theorem, we know that, (x - a) is a factor of p(x) only if p(a) = 0

On comparing (x + 2) = x - (-2) with (x - a), we get,

• a = -2

Now as (x + 2) is a factor of p(x), hence,

→ p(-2) = 0

→ p(-2) = (-2)⁴ + (-2)³ - 14(-2) - k = 0

→ 16 - 8 + 28 - k = 0

→ k = 36 Ans.

Extra knowledge:-

For a polynomial ax² + bx + c, having roots x₁ and x₂,

• Sum of roots = x₁ + x₂ = -b/a
• Product of roots = x₁x₂ = c/a