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.)
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)=3x2−5x+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.