Answer :
Certainly! Let's go through the process of synthetic division step by step using the given numbers.
We are given the polynomial [tex]\( f(x) = x^3 + 6x^2 - 7x - 60 \)[/tex].
We will test a potential root, which is [tex]\( x = -5 \)[/tex], using synthetic division.
1. Write down the coefficients:
The coefficients from the polynomial are: [tex]\( 1, 6, -7, -60 \)[/tex].
2. Set up the synthetic division:
You will place the potential root [tex]\(-5\)[/tex] to the left, then draw a vertical bar. Write the coefficients of the polynomial to the right of this bar in order:
```
-5 | 1 6 -7 -60
```
3. Bring down the leading coefficient:
The leading coefficient, which is [tex]\(1\)[/tex], is written below the line:
```
| 1 6 -7 -60
----------------
| 1
```
4. Multiply and add down:
- Multiply the leading coefficient by the root: [tex]\( -5 \times 1 = -5 \)[/tex].
- Add this product to the next coefficient: [tex]\( 6 + (-5) = 1 \)[/tex].
Update the row:
```
| 1 6 -7 -60
----------------
| 1 1
```
- Multiply the root by the new number: [tex]\( -5 \times 1 = -5 \)[/tex].
- Add to the next coefficient: [tex]\( -7 + (-5) = -12 \)[/tex].
Update the row:
```
| 1 6 -7 -60
----------------
| 1 1 -12
```
- Multiply the root by the new number: [tex]\( -5 \times (-12) = 60 \)[/tex].
- Add to the next coefficient: [tex]\( -60 + 60 = 0 \)[/tex].
Update the row:
```
| 1 6 -7 -60
----------------
| 1 1 -12 0
```
5. Interpreting the result:
The bottom row now reads: [tex]\(1, 1, -12, 0\)[/tex].
- The first three numbers [tex]\(1, 1, -12\)[/tex] are the coefficients of the quotient polynomial [tex]\( x^2 + x - 12 \)[/tex].
- The final number, [tex]\(0\)[/tex], is the remainder.
Since the remainder is 0, [tex]\(-5\)[/tex] is indeed a root of the polynomial. The division is complete.
So, the numbers that complete the synthetic division are [tex]\(1, 1, -12, 0\)[/tex].
We are given the polynomial [tex]\( f(x) = x^3 + 6x^2 - 7x - 60 \)[/tex].
We will test a potential root, which is [tex]\( x = -5 \)[/tex], using synthetic division.
1. Write down the coefficients:
The coefficients from the polynomial are: [tex]\( 1, 6, -7, -60 \)[/tex].
2. Set up the synthetic division:
You will place the potential root [tex]\(-5\)[/tex] to the left, then draw a vertical bar. Write the coefficients of the polynomial to the right of this bar in order:
```
-5 | 1 6 -7 -60
```
3. Bring down the leading coefficient:
The leading coefficient, which is [tex]\(1\)[/tex], is written below the line:
```
| 1 6 -7 -60
----------------
| 1
```
4. Multiply and add down:
- Multiply the leading coefficient by the root: [tex]\( -5 \times 1 = -5 \)[/tex].
- Add this product to the next coefficient: [tex]\( 6 + (-5) = 1 \)[/tex].
Update the row:
```
| 1 6 -7 -60
----------------
| 1 1
```
- Multiply the root by the new number: [tex]\( -5 \times 1 = -5 \)[/tex].
- Add to the next coefficient: [tex]\( -7 + (-5) = -12 \)[/tex].
Update the row:
```
| 1 6 -7 -60
----------------
| 1 1 -12
```
- Multiply the root by the new number: [tex]\( -5 \times (-12) = 60 \)[/tex].
- Add to the next coefficient: [tex]\( -60 + 60 = 0 \)[/tex].
Update the row:
```
| 1 6 -7 -60
----------------
| 1 1 -12 0
```
5. Interpreting the result:
The bottom row now reads: [tex]\(1, 1, -12, 0\)[/tex].
- The first three numbers [tex]\(1, 1, -12\)[/tex] are the coefficients of the quotient polynomial [tex]\( x^2 + x - 12 \)[/tex].
- The final number, [tex]\(0\)[/tex], is the remainder.
Since the remainder is 0, [tex]\(-5\)[/tex] is indeed a root of the polynomial. The division is complete.
So, the numbers that complete the synthetic division are [tex]\(1, 1, -12, 0\)[/tex].
