Answer: (a) \(a=17\), \(b=-8\); (b) \(-8\); (c) \(34\).

Answer: (a) \(a=17\), \(b=-8\); (b) \(-8\); (c) \(34\).

Since \(p(x)\) is divisible by \(2x-1\), \(x=\frac12\) is a root. Hence

\(p\left(\frac12\right)=0.\)

So

This gives

\(\frac34+\frac{a}{4}+3+b=0.\)

Multiplying by \(4\),

\(a+4b+15=0.\)

When divided by \(x-2\), the remainder is \(p(2)\), so

\(p(2)=120.\)

Thus

\(48+4a+12+b=120,\)

which simplifies to

\(4a+b=60.\)

Solving

\(a+4b=-15,\qquad 4a+b=60\)

gives

\(a=17,\qquad b=-8.\)

The remainder when \(p(x)\) is divided by \(x\) is \(p(0)\), which is \(b\). Hence the remainder is \(-8\).

Now

\(p'(x)=18x^2+34x+6,\)

and

\(p''(x)=36x+34.\)

Therefore

\(p''(0)=34.\)