Solution

Elementary Linear algebra

Chapter 1.9

Exercise 17

a)

Let $p(x)=a+bx+cx^2$ be polynomial that passes through points $(0,1)$ and $(1,2)$. Then

$$\begin{align*} 1 &= p(0) = a+b\cdot 0+ c\cdot 0^2= a\\ 2 &= p(1) = a+b\cdot 1 + c\cdot 1^2 = a+b+c \end{align*}$$

Thus, for any $t\in\mathbb{R}$ we have

Answer

p(x)=1+tx+(1−t)x2 , ∀t∈R

$$\begin{align*} a &= 1 \\ b &= t \\ c &= 1-t \end{align*}$$