Proof that p=r when x-2 and x-1/2 are factors
Explanation:
- Let's assume that both x-2 and x-1/2 are factors of the polynomial px²+5x+r. This means that when x=2 and x=-1/2, the polynomial will equal to zero.
- Substituting x=2 into the polynomial px²+5x+r gives us p(2)²+5(2)+r = 4p+10+r.
- Since x-2 is a factor, the polynomial will equal zero when x=2, so we have 4p+10+r=0. This simplifies to 4p+r=-10. This is our first equation.
- Substituting x=-1/2 into the polynomial px²+5x+r gives us p(-1/2)²+5(-1/2)+r = p/4-5/2+r.
- Since x-1/2 is a factor, the polynomial will equal zero when x=-1/2, so we have p/4-5/2+r=0. This simplifies to p+r=10/2=5. This is our second equation.
- Now we have a system of two equations: 4p+r=-10 and p+r=5.
- Solving this system of equations, we get p=5 and r=5.
- Therefore, when both x-2 and x-1/2 are factors of px²+5x+r, we have shown that p=r.