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.)

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.)

Transcript text: For the polynomial $\mathrm{P}(\mathrm{x})=3 \mathrm{x}^{2}-5 \mathrm{x}+3$ and $\mathrm{c}=1$, find $\mathrm{P}(\mathrm{c})$ by (a) direct substitution and (b) the remainder theorem. a. Find $P(1)$ by direct substitution. $P(1)=$ $\square$ (Type an integer.)

Solution

Solution Steps

Step 1: Direct Substitution

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

\[ P(1) = 3(1)^2 - 5(1) + 3 \]

\[ P(1) = 3(1) - 5(1) + 3 \]

\[ P(1) = 3 - 5 + 3 \]

\[ P(1) = 1 \]

Step 2: Remainder Theorem

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 $.