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.)
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Solution

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Solution Steps

Step 1: Direct Substitution

To find P(1) P(1) by direct substitution, substitute x=1 x = 1 into the polynomial P(x)=3x25x+3 P(x) = 3x^2 - 5x + 3 :

P(1)=3(1)25(1)+3 P(1) = 3(1)^2 - 5(1) + 3

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

P(1)=35+3 P(1) = 3 - 5 + 3

P(1)=1 P(1) = 1

Step 2: Remainder Theorem

The remainder theorem states that if a polynomial P(x) P(x) is divided by xc x - c , the remainder is P(c) P(c) . Here, c=1 c = 1 , so the remainder when P(x) P(x) is divided by x1 x - 1 is P(1) P(1) .

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

Final Answer

P(1)=1 \boxed{P(1) = 1}

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