Exercise 69 Page 186

Let's start by reviewing what the The Remainder Theorem states.

If a polynomial function $f(x)$ is divided by $x-k,$ then the remainder is $r=f(k).$

The Remainder Theorem tells us that we can find the value of the polynomial function at $x=k$ by dividing it by the binomial $x-k.$ This allows us to evaluate the polynomials using synthetic division. Let's take a look at the exercise's function. Since we want to find $g(7),$ we could either evaluate it by directly substituting $x=7$ or by using synthetic division to divide $g(x)$ by $x-7.$ We will show both methods to compare them.

Evaluating $g(7)$ by Direct Substitution

Let's evaluate the function by substitute $x=7$ in $g(x).$ Notice that these calculations are not that easy if we cannot use a calculator.

Evaluating $g(7)$ by Using Synthetic Division

According to the Remainder Theorem, $g(7)$ is equal to the remainder of the division of $g(x)$ by $x-7.$ Let's calculate this quotient using synthetic division. Since the remainder is $0,$ we can know that $g(7)=0.$ Also, according to the Factor Theorem we can conclude that $x-7$ is a factor of the polynomial.

A polynomial $f(x)$ has a factor $x-k$ if and only if $f(k)=0.$

Conclusions

As we can see, evaluating the function by using synthetic division is easier in this case, since the calculations involved are simpler than those needed for evaluating by directly substituting $x=7.$