To determine the value of b when x² is a factor of the polynomial x² + bx + b, we start by recognizing what it means for x² to be a factor. If x² is a factor of the expression, then we can express it as:
x² + bx + b = x²(q(x))
for some polynomial q(x). We can assume that q(x) is a linear polynomial, since x² is a quadratic polynomial and dividing it could result in a linear polynomial.
Let’s rewrite the polynomial in the form:
x² + bx + b = x²(1) + 0(x) + b
We need to find conditions under which x² divides the polynomial evenly.
Since x² is a factor, the remaining factor must be a polynomial that evaluates to zero when x = 0. This means that b must be equal to zero, otherwise the constant term would prevent x² from being a factor.
Let’s set b = 0:
x² + 0*x + 0 = x² (which shows that x² is indeed a factor).
Thus, the condition x² divides x² + bx + b leads to the conclusion that:
b = 0
In summary, the only value of b that allows x² to be a factor of the polynomial x² + bx + b is when b is equal to zero.