Question

15.
If f(x) = x² +5x+p and g(x) = x+3x+q have a common factor, then
i) Find the common factor
ii) Show that (p-q)2 = 2(3p-59)




plese help me only (i) one​

Answer 1

• Zero of Equations

If we substitute the common solution,

both equations will show 0.

• Factor Theorem

If one solution is α, the factor is always x-α.

From the question.

There will be a common zero that satisfies the equations.

• $\sf{f(\alpha )=\alpha ^2+5\alpha +p=0}$ ...(1)

• $\sf{g(\alpha )=\alpha ^2+3\alpha +q=0}$ ...(2)

After we solve the system equation we obtain

$\sf{2\alpha+p-q =0}$

∴$\sf{\alpha =-\dfrac{p-q}{2} }$

We said one common zero was α

Therefore the common factor is

∴$\sf{x-\alpha =x+\dfrac{p-q}{2}}$

For your information.

We get common factor x-α

because we used factor theorem.