Answer :

We want to divide
$$
P(x) = x^4 - 7x^3 + 5x - 45
$$
by
$$
D(x) = x - 7.
$$

### Step 1. Write the Polynomial with All Terms

Notice that the polynomial is missing the $x^2$ term. We rewrite it with a zero coefficient for $x^2$:

$$
P(x) = 1x^4 - 7x^3 + 0x^2 + 5x - 45.
$$

### Step 2. Set Up Synthetic Division

Since $D(x) = x - 7$, we set $x=7$ for synthetic division. Write down the coefficients:

$$
1, \quad -7, \quad 0, \quad 5, \quad -45.
$$

### Step 3. Perform Synthetic Division

1. **Bring down the first coefficient:**
The first coefficient is $1$.
$$\text{Carry over: } 1$$

2. **Multiply and add to the next coefficient:**
Multiply the carry $1$ by $7$:
$$1 \cdot 7 = 7.$$
Add this to the next coefficient $-7$:
$$-7 + 7 = 0.$$

3. **Repeat the process:**
Multiply the new carry $0$ by $7$:
$$0 \cdot 7 = 0.$$
Add it to the next coefficient $0$:
$$0 + 0 = 0.$$

4. **Next step:**
Multiply the new carry $0$ by $7$:
$$0 \cdot 7 = 0.$$
Add to the next coefficient $5$:
$$5 + 0 = 5.$$

5. **Final step for the remainder:**
Multiply the carry $5$ by $7$:
$$5 \cdot 7 = 35.$$
Add to the last coefficient $-45$:
$$-45 + 35 = -10.$$

This final number is the remainder.

### Step 4. Write the Result

The numbers obtained (except for the remainder) form the coefficients of the quotient starting from the highest degree. The coefficients are:
$$
1,\quad 0,\quad 0,\quad 5.
$$

This corresponds to the quotient polynomial:
$$
Q(x) = 1x^3 + 0x^2 + 0x + 5 = x^3 + 5.
$$

The remainder is:
$$
R = -10.
$$

### Final Answer

Dividing
$$
x^4 - 7x^3 + 5x - 45
$$
by
$$
x - 7
$$
gives the quotient
$$
x^3 + 5
$$
with remainder
$$
-10.
$$

Thus, we can write:
$$
\frac{x^4 - 7x^3 + 5x - 45}{x - 7} = x^3 + 5 \quad \text{with remainder} \quad -10.
$$

Other Questions