Property 12
Example
| \(P(x)\) | \(x-1\) |
| _ |
| \(3\) |
| \(P(x)\) | \(x+1\) |
| _ |
| \(7\) |
| \(P(x)\) | \(x^2-1\) |
| _ |
| \(K(x)\) |
\(\Rightarrow K(x)=?\)
Answer
\(P(1)=3\)
\(P(-1)=7\)
\(P(x)=(x^2-1)\cdot B(x)+K(x)\)
Remainder is a linear polynomial
\(K(x)=ax+b\)
\(P(1)=a+b=3\)
\(P(-1)=-a+b=7\)
\(a=-2 \qquad b=5\)
\(K(x)=-2x+5\)
1.
| \(P(x)\) | \(x-2\) |
| _ |
| \(3\) |
| \(P(x)\) | \(x-1\) |
| _ |
| \(2\) |
| \(P(x)\) | \(x^2-3x+2\) |
| _ |
| \(K(x)\) |
\(\Rightarrow K(x)=?\)
2.
| \(P(x)\) | \(x-1\) |
| _ |
| \(-1\) |
| \(P(x)\) | \(x+2\) |
| _ |
| \(2\) |
| \(P(x)\) | \(x^2+x-2\) |
| _ |
| \(K(x)\) |
\(\Rightarrow K(x)=?\)
3.
| \(P(x)\) | \(x+2\) |
| _ |
| \(4\) |
| \(P(x)\) | \(x+1\) |
| _ |
| \(2\) |
| \(P(x)\) | \(x^2+3x+2\) |
| _ |
| \(K(x)\) |
\(\Rightarrow K(x)=?\)
4.
| \(P(x)\) | \(x-2\) |
| _ |
| \(0\) |
| \(P(x)\) | \(x+2\) |
| _ |
| \(-2\) |
| \(P(x)\) | \(x^2-4\) |
| _ |
| \(K(x)\) |
\(\Rightarrow K(x)=?\)
5.
| \(x^2+ax+b+3\) | \((x-1)(x+2)\) |
| _ |
| \(0\) |
\(\Rightarrow a+b=?\)
6.
| \(x^3+ax^2-bx+2\) | \((x-2)(x+3)\) |
| _ |
| \(2\) |
\(\Rightarrow a+b=?\)
7.
| \(x^3+ax^2+3x+b\) | \((x-2)(x-1)\) |
| _ |
| \(0\) |
\(\Rightarrow a=?\)
8.
| \(x^3+ax^2+bx+2\) | \((x-2)(x+1)\) |
| _ |
| \(0\) |
\(\Rightarrow a+b=?\)