Property 11
Example
| \(P(x)=2x^3-4x^2+5x-7\) | \(x^2+1\) |
| _ | \(B(x)\) |
| \(K(x)\) |
\(\Rightarrow K(x)=?\)
Answer
Without using a classic division method remainder can be found.
\(x^2+1=0 \Rightarrow x^2=-1\)
In polynomial \(-1\) is instead of \(x^2\).
\(P(x)=2x^3-4x^2+5x-7\)
\(=2x^2\cdot x-4x^2+5x-7\)
\(=2(-1)\cdot x-4(-1)+5x-7\)
\(=-2x+4+5x-7\)
\(=3x-3\)
\(\Rightarrow K(x)=3x-3\)
1.
| \(x^4-2x^2+4\) | \(x^2-1\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
2.
| \(2x^6-2x^3+6\) | \(x^3-2\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
3.
| \(x^{12}-4x^8+6x^4\) | \(x^4-1\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
4.
| \(2x^6-3x^3+a\) | \(x^3+2\) |
| _ |
| \(0\) |
\(\Rightarrow a=?\)
5.
| \(4x^3-2x^2+1\) | \(x^2-1\) |
| _ |
| \(K(x)\) |
\(\Rightarrow K(x)=?\)
6.
| \(x^5-2x^3+x^2-3\) | \(x^2-2\) |
| _ |
| \(K\) |
\(\Rightarrow K=?\)
7.
| \(x^4-2x^3+ax\) | \(x^3-1\) |
| _ |
| \(3x-2\) |
\(\Rightarrow a=?\)
8.
| \(x^5-3x^3+x^2+a\) | \(x^3+2\) |
| _ |
| \(-x^2+7\) |
\(\Rightarrow a=?\)
9.
| \(x^3+ax^2+b+2\) | \(x^2-x\) |
| _ |
| \(3x+3\) |
\(\Rightarrow a+b=?\)
10.
| \(x^2+ax+b+5\) | \(x^2-3x+2\) |
| _ |
| \(0\) |
\(\Rightarrow a=?\)