If the polynomial ax³ + bx² + c is divisible by x² + bx + c, what can we say about the relationship between a and b?

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To determine the relationship between the coefficients a and b in the polynomial ax³ + bx² + c when it is divisible by x² + bx + c, we first need to utilize the principles of polynomial division.

Since ax³ + bx² + c is divisible by x² + bx + c, it implies that there exists some polynomial Q(x) such that:

ax³ + bx² + c = (x² + bx + c) * Q(x)

Given that x² + bx + c is a quadratic polynomial, we can express the quotient Q(x) as:

Q(x) = mx + n

where m and n are constants. The degree of the resultant polynomial ax³ + bx² + c suggests that Q(x) must be a linear polynomial.

Expanding this expression yields:

ax³ + bx² + c = (x² + bx + c)(mx + n)

Next, we perform the multiplication:

ax³ + bx² + c = mx³ + (mb + n)x² + (mc + nb)x + nc

From the coefficients of like powers of x, we can equate:

  • For x³: m = a
  • For x²: mb + n = b
  • For x: mc + nb = 0
  • For the constant term: nc = c

From m = a, we know that:

mb + n = ab
ab + n = b

Rearranging gives us:

n = b - ab

Continuing with nc = c, if we assume c ≠ 0, we can derive that:

n = rac{c}{c}

Next, substituting back gives valuable insight about the relationship:

For all these equations to be satisfied consistently, particularly the terms that involve n, we get:

ab + rac{c}{c} = b

This leads to the conclusion:

  • ab must equal 0 or 1.

This indicates that for the polynomial ax³ + bx² + c to maintain divisibility by x² + bx + c, one of the coefficients, either a or b, must appropriately align to make the divisibility consistent. Therefore, in conclusion, under the conditions of polynomial division, one can derive that:

ab ≠ 1

That means either a = 0 or b = 1.

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