Answer: choice D
x = -4 is not a zero of the polynomial
quotient = x-4
remainder = -153
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Work Shown:
q = quotient = ax+b
r = remainder
r = p(-4), aka the function output of p(x) when the value x = -4 is plugged in.
p(x) = x^2 - 169
p(-4) = (-4)^2 - 169
p(-4) = 16 - 169
p(-4) = -153 is the remainder, so r = -153.
Since the remainder is nonzero, this means x = -4 is not a root or zero or x intercept of p(x). This also means x+4 is not a factor of p(x).
p(x)/(x+4) = q + r/(x+4)
p(x) = q(x+4) + r
x^2 - 169 = q(x+4) - 153
x^2 - 169 = (ax+b)(x+4) - 153 ... replace 'q' with ax+b
x^2 + 0x - 169 = ax^2 + 4ax + bx + 4b - 153
x^2 + 0x - 169 = ax^2 + (4a+b)x + (4b-153)
We see that x^2 on the left side matches with ax^2 on the right side, so a = 1.
The middle terms 0x and (4a+b)x match up, so 0 = 4a+b. Use a = 1 to find that b = -4. So the quotient is q = ax+b = 1x+(-4) = x-4
You can also use polynomial long division to achieve the same goal. See the attached image below.