(j) When the function $$ 4 x^{2}+2 q x+r+q $$ is divided by $$ 2 x-1 $$ and $$ x+2 $$ the remainders are -1 and 9 respectively. Determine the values of $$ q $$ and $$ r $$.

Question

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We are given the function

$$
f(x)=4x^2+2qx+r+q
$$

and the conditions:

1. When divided by $$2x-1$$, the remainder is $$-1$$. Since the remainder theorem tells us to substitute the value where the divisor equals zero, we have:
   $$
   2x-1=0 \implies x=\frac{1}{2}
   $$
   Thus:
   $$
   f\left(\frac{1}{2}\right) = -1
   $$

2. When divided by $$x+2$$, the remainder is $$9$$. Again, setting the divisor equal to zero:
   $$
   x+2=0 \implies x=-2
   $$
   Thus:
   $$
   f(-2)=9
   $$

**Step 1. Evaluate $$f\left(\frac{1}{2}\right)$$:**

Substitute $$x=\frac{1}{2}$$ into $$f(x)$$:

$$
f\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^2 + 2q\left(\frac{1}{2}\right) + r + q
$$

Calculate each term:

$$
4\left(\frac{1}{2}\right)^2 = 4\left(\frac{1}{4}\right) = 1
$$
$$
2q\left(\frac{1}{2}\right) = q
$$

Thus, we have:

$$
f\left(\frac{1}{2}\right) = 1 + q + r + q = 1 + r + 2q
$$

Setting it equal to $$-1$$:

$$
1 + r + 2q = -1
$$

Subtract 1 from both sides:

$$
r + 2q = -2 \quad \text{(Equation 1)}
$$

**Step 2. Evaluate $$f(-2)$$:**

Substitute $$x=-2$$ into $$f(x)$$:

$$
f(-2)= 4(-2)^2 + 2q(-2) + r + q
$$

Calculate each term:

$$
4(-2)^2 = 4(4) = 16
$$
$$
2q(-2)= -4q
$$

Thus:

$$
f(-2)= 16 - 4q + r + q = 16 + r - 3q
$$

Setting it equal to $$9$$:

$$
16 + r - 3q = 9
$$

Subtract 16 from both sides:

$$
r - 3q = 9 - 16 = -7 \quad \text{(Equation 2)}
$$

**Step 3. Solve the system of equations:**

We have the system:
$$
\begin{cases}
r + 2q = -2 \quad \text{(1)}\[6pt]
r - 3q = -7 \quad \text{(2)}
\end{cases}
$$

Subtract Equation (2) from Equation (1):

$$
(r + 2q) - (r - 3q) = -2 - (-7)
$$
$$
2q + 3q = 5
$$
$$
5q = 5 \implies q = 1
$$

Substitute $$q=1$$ into Equation (1):

$$
r + 2(1) = -2
$$
$$
r + 2 = -2 \implies r = -4
$$

**Final Answer:**

$$
q=1 \quad \text{and} \quad r=-4
$$
$$ q = 1 $$ and $$ r = -4 $$.

(j) When the function is divided by and the remainders
are -1 and 9 respectively. Determine the values of and .

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(j) When the function is divided by and the remainders are -1 and 9 respectively. Determine the values of and .