Multiple choice

A quadratic polynomial is exactly divisible by (x + 1) & (x + 2) and leaves the remainder 4 after division by (x + 3) then that polynomial is

  1. $x^2 +6x+ 4$
  2. $2x^2 + 6x + 4$
  3. $2x^2 +6x-4$
  4. $x^2 +6x-4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If the polynomial P(x) is divisible by (x+1) and (x+2), then P(x) = a(x+1)(x+2). Given P(-3) = 4, we have a(-2)(-1) = 4, so 2a = 4, a = 2. Thus, P(x) = 2(x^2 + 3x + 2) = 2x^2 + 6x + 4.

AI explanation

Using the remainder theorem, since (x + 1) and (x + 2) are exact divisors, the polynomial evaluated at -1 and -2 must equal 0. Testing the first option for the polynomial p(x) = x^2 + 6x + 4 gives p(-1) = -1 and p(-2) = -4, so it is incorrect. Testing the second option, p(x) = 2x^2 + 6x + 4, gives p(-1) = 0 and p(-2) = 0, satisfying the divisibility condition. Checking the remainder condition for this polynomial at x = -3 yields 2(-3)^2 + 6(-3) + 4 = 18 - 18 + 4 = 4, proving the polynomial is 2x^2 + 6x + 4.

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