Property 6
Equality of the given polynomial \(P(x)\) is chosen properly according to the given equation.
If it's linear polynomial then \(P(x)=ax+b\)
If it's quadratic polynomial then \(P(x)=ax^2+bx+c\)
1.
\(2P(x)+P(-x)=x+9\)
\(\Rightarrow P(x)=?\)
2.
\(P(x+1)+P(x-1)=4x+6\)
\(\Rightarrow P(x)=?\)
3.
\(P(x+2)+P(x-2)=2x+6\)
\(\Rightarrow P(2)=?\)
4.
\(P(x)+P(2x)=6x+8\)
\(\Rightarrow P(x)=?\)
5.
\(P(x)+P(x+1)=6x+1\)
\(\Rightarrow P(1)=?\)
6.
\(P(x)\) positive coefficient polynomial
\(P(x)\cdot P(2x)=2x^2+6x+4\)
\(\Rightarrow P(x)=?\)
7.
\(P(x)\) positive coefficient polynomial
\(P(x)\cdot P(-x)=1-4x^2\)
\(\Rightarrow P(2)=?\)
8.
\(P(x-2)-2P(x)=-x^2-6x+5\)
\(\Rightarrow P(x)=?\)
Property 7
If there is an unknown term in equations it can be determined by equalizing the factor next to the \(P(x)\) polynomial to zero. Then it can be simplified.
1.
\((x-1)\cdot P(x)=x^2-kx-5\)
\(\Rightarrow k=?\)
2.
\((x-2)\cdot P(x)=ax^2-3x+2\)
\(\Rightarrow a=?\)
3.
\((x+1)\cdot P(x)=x^3-2a\)
\(\Rightarrow a=?\)
4.
\((x+2)\cdot P(x)=x^2-2kx\)
\(\Rightarrow P(3)=?\)
5.
\((x-3)\cdot P(x)=x^2-5x+k\)
\(\Rightarrow P(1)=?\)
6.
\((x+2)\cdot P(x)=x^2+5x+k\)
\(\Rightarrow P(-2)=?\)
7.
\((x+1)\cdot P(x)=x^2+3x+k\)
\(\Rightarrow P(-1)=?\)