Find a monic quadratic polynomial f(x) such that the remainder when f(x) is divided by x-1 is 2 and the remainder when f(x) is divided by x-3 is 4. Give your answer in the form .
+26367 Find a monic quadratic polynomial f(x) such that the remainder when f(x) is divided by x-1 is 2
and the remainder when f(x) is divided by x-3 is 4.
\(\begin{array}{|rcll|} \hline \mathbf{f(x)} &=& \mathbf{q_1(x)(x-1) + 2} \quad &| \quad x = 1 \\ f(1) &=& q_1(x)*0 + 2 \\ \mathbf{f(1)} &=& \mathbf{2} \\\\ \mathbf{f(x)} &=& \mathbf{ q_2(x)(x-3) + 4 } \quad &| \quad x = 3 \\ f(3) &=& q_2(x)*0 + 4 \\ \mathbf{f(3)} &=& \mathbf{4} \\ \hline \end{array}\)
Monic quadratic polynomial \(f(x) = x^2+ax+b\)
\(\begin{array}{|rcll|} \hline \mathbf{f(x)} &=& \mathbf{x^2+ax+b} \\\\ f(1) &=& 1^2+a*1 + b \quad & | \quad f(1) = 2 \\ 2 &=& 1+a+b \\ 1 &=& a+b \\ \mathbf{b} &=& \mathbf{1-a} \\\\ f(3) &=& 3^2+a*3 + b \quad & | \quad f(3) = 4 \\ 4 &=& 9+3a + b \quad &| \quad b=1-a \\ 4 &=& 9+3a + 1-a \\ 4 &=& 10+2a \\ -6 &=& 2a \\ a &=& -\dfrac{6}{2} \\ \mathbf{a} &=& \mathbf{-3} \\\\ \mathbf{b} &=& \mathbf{1-a} \\ b &=& 1-(-3) \\ b &=& 1+3 \\ \mathbf{b} &=& \mathbf{4} \\ \hline \end{array} \)
\(\mathbf{f(x) = x^2-3x+4 }\)
