Final answer:
The provided quotient of the polynomial division indicates that x+4 is not a factor of p(x) since there's a remainder term (-144/(x+4)). This implies that -4 is not a root of p(x). Understanding the remainder's role is crucial in polynomial division.
Step-by-step explanation:
If a polynomial p(x) is divided by x+4, the result given is x2−7x+34−144/(x+4). To evaluate this, one must understand the process of polynomial long division, which is similar in structure to long division with numbers. From the provided quotient, one must recognize that the final term, −144/(x+4), is the remainder of the division.
It follows that p(x) can be expressed as (x+4)(x2−7x+34) - 144. If we were to multiply this out, we would receive the original polynomial p(x). Given this information, there are certain implications regarding the original polynomial and the divisor (x+4). Firstly, since there's a remainder, x+4 is not a factor of the polynomial p(x). Secondly, the value −4 is not a root of p(x).
To further elucidate, if p(x) were exactly divisible by x+4, the remainder would be zero, and −4 would be a root of the polynomial, meaning that p(−4) would equal zero. Thus, seeing a remainder when dividing a polynomial by a linear divisor indicates that the divisor is not a perfect factor and the negated constant term (in this case, −4) is not a zero of the polynomial.