f(x)=px^3 + 6x^2 + 12x + q

given that the remainder when f(x) is divided by (x - 1) is equal to the remainder when f(x) is divided by (2x + 1),

(a) find the value of p.

given also that q = 3, and p has the value found in part (a)

(b) find the value of the remainder.

my working out for (a) is

f(1)=p(1)^3 + 6(1)^2 + 12(1) + q

= p + 6 + 12 + q

= p + 18 = q

then I get stuck. Please help.

Do the same thing you did before using -1/2 for x. (2x+1=0 and solve for x) Then you will have two equations and two unknowns and you can solve for p

Like simultaneous equations?

p + 18 + q = 1/8p - 4.5 +q

I get -25.71..... Is this right?

To find the value of p in part (a), you correctly set up the equation f(1) = p + 18 = q. However, you made a small mistake when calculating f(1).

Let's go through the correct calculations step by step.

1. Start with the given equation f(x) = px^3 + 6x^2 + 12x + q.

2. Calculate the remainder when f(x) is divided by (x - 1):

To find the remainder when f(x) is divided by (x - 1), substitute x = 1 into f(x) and evaluate:

f(1) = p(1)^3 + 6(1)^2 + 12(1) + q
= p + 6 + 12 + q
= p + 18 + q

3. Calculate the remainder when f(x) is divided by (2x + 1):

To find the remainder when f(x) is divided by (2x + 1), substitute x = -1/2 into f(x) and evaluate:

f(-1/2) = p(-1/2)^3 + 6(-1/2)^2 + 12(-1/2) + q
= -p/8 + 3/4 - 6 + q
= -p/8 - 23/4 + q

4. Since the remainders when f(x) is divided by (x - 1) and (2x + 1) are equal, we have:

p + 18 + q = -p/8 - 23/4 + q

5. Simplify the equation:

Multiply both sides of the equation by 8 to eliminate the fractions:

8(p + 18 + q) = -(p + 23) + 8q

8p + 144 + 8q = -p - 23 + 8q

9p = -167

6. Solve for p:

Divide both sides of the equation by 9:

p = -167/9

So, the value of p is -167/9.

For part (b), let's substitute the value of q from the given information, q = 3, and the value of p we found, p = -167/9, into the original equation f(x):

f(x) = px^3 + 6x^2 + 12x + q
= (-167/9)x^3 + 6x^2 + 12x + 3

To find the remainder when f(x) is divided by (x - 1), substitute x = 1 into f(x) and evaluate:

f(1) = (-167/9)(1)^3 + 6(1)^2 + 12(1) + 3
= -167/9 + 6 + 12 + 3
= -167/9 + 75/9
= -92/9

So, the value of the remainder when f(x) is divided by (x - 1) is -92/9.