Questions: For the polynomial P(x)=3x^2-5x+3 and c=1, find P(c) by (a) direct substitution and (b) the remainder theorem. a. Find P(1) by direct substitution. P(1)= (Type an integer.)

Solution Steps

To find $P(1)$ by direct substitution, substitute $x = 1$ into the polynomial $P(x) = 3x^2 - 5x + 3$:

$$P(1) = 3(1)^2 - 5(1) + 3$$

$$P(1) = 3(1) - 5(1) + 3$$

$$P(1) = 3 - 5 + 3$$

$$P(1) = 1$$

The remainder theorem states that if a polynomial $P(x)$ is divided by $x - c$, the remainder is $P(c)$. Here, $c = 1$, so the remainder when $P(x)$ is divided by $x - 1$ is $P(1)$.

From Step 1, we already found that $P(1) = 1$. Therefore, by the remainder theorem, the remainder is also $1$.

Final Answer