If #h(x)=f(x)g(x)# and #f(x)=2x+5#, how do you determine g(x) given #h(x)=2xsqrtx+5sqrtx#?

Question

Serenity Johnson · Accepted Answer

#g(x)=sqrtx.# #h(x)=f(x)g(x)# #rArr 2xsqrtx+5sqrtx=(2x+5)g(x)# #rArr (2x+5)sqrtx=(2x+5)g(x)# #"Dividing by "(2x+5)!=0, sqrtx=g(x)# Thus, #g(x)=sqrtx. (x!=-5/2).#

James Allen · Answer

undefined To determine $ g(x) $, given $ h(x) = f(x)g(x) $ and $ f(x) = 2x + 5 $, you can follow these steps:

1. Rewrite the expression for $ h(x) $ by substituting the given expression for $ f(x) $.
2. Compare the resulting expression for $ h(x) $ with the given expression for $ h(x) $.
3. Identify the corresponding terms between the two expressions to determine $ g(x) $.

Given that $ h(x) = 2x\sqrt{x} + 5\sqrt{x} $ and $ f(x) = 2x + 5 $, we substitute $ f(x) $ into the expression for $ h(x) $:

$$ h(x) = (2x + 5) \cdot g(x) $$

Now, we compare the given expression for $ h(x) $ with the rewritten expression:

$$ 2x\sqrt{x} + 5\sqrt{x} = (2x + 5) \cdot g(x) $$

By comparing corresponding terms, we see that $ g(x) $ must contain $ \sqrt{x} $, and the coefficient of $ \sqrt{x} $ in $ g(x) $ is the sum of the coefficients of $ 2x $ and $ 5 $ in the expression $ h(x) $, which is $ 2 + 5 = 7 $.

Therefore, $ g(x) = 7\sqrt{x} $.