Answer :
Sure! Let's go through the process of using synthetic division to divide the polynomial [tex]\( x^4 + 16x^3 + 67x^2 + 63x - 70 \)[/tex] by [tex]\( x + 10 \)[/tex].
### Step-by-Step Solution:
1. Write Down the Coefficients:
The polynomial [tex]\( x^4 + 16x^3 + 67x^2 + 63x - 70 \)[/tex] has the coefficients: 1, 16, 67, 63, and -70.
2. Set Up for Synthetic Division:
Since we are dividing by [tex]\( x + 10 \)[/tex], we use the number [tex]\(-10\)[/tex] for the division process.
3. Perform the Synthetic Division:
- Write down [tex]\(-10\)[/tex], the number we are using for division.
- Bring down the first coefficient (1):
- This becomes the first number in our result row.
- Multiply and Add:
- Multiply the number in the result row (initially just a copy of the first coefficient, which is 1) by [tex]\(-10\)[/tex].
- Write the result below the next coefficient.
- Add this number to the next coefficient.
- Repeat the Process:
- Continue this process for each coefficient:
- Multiply the result from the previous step by [tex]\(-10\)[/tex].
- Add this product to the next coefficient in the sequence.
- Fill in the Result Row:
- Perform these steps until you've processed all coefficients.
4. Obtain the Quotient and Remainder:
- The numbers in the result row except for the last number are the coefficients of the quotient polynomial.
- The last number is the remainder.
### Final Result:
After performing synthetic division, the quotient is the polynomial:
[tex]\[ x^3 + 6x^2 + 7x - 7 \][/tex]
The remainder is:
[tex]\[ 0 \][/tex]
Thus, the division of [tex]\( x^4 + 16x^3 + 67x^2 + 63x - 70 \)[/tex] by [tex]\( x + 10 \)[/tex] results in:
[tex]\[ x^3 + 6x^2 + 7x - 7 \][/tex] with a remainder of 0.
This zero remainder confirms that [tex]\( x+10 \)[/tex] is a factor of the given polynomial.
### Step-by-Step Solution:
1. Write Down the Coefficients:
The polynomial [tex]\( x^4 + 16x^3 + 67x^2 + 63x - 70 \)[/tex] has the coefficients: 1, 16, 67, 63, and -70.
2. Set Up for Synthetic Division:
Since we are dividing by [tex]\( x + 10 \)[/tex], we use the number [tex]\(-10\)[/tex] for the division process.
3. Perform the Synthetic Division:
- Write down [tex]\(-10\)[/tex], the number we are using for division.
- Bring down the first coefficient (1):
- This becomes the first number in our result row.
- Multiply and Add:
- Multiply the number in the result row (initially just a copy of the first coefficient, which is 1) by [tex]\(-10\)[/tex].
- Write the result below the next coefficient.
- Add this number to the next coefficient.
- Repeat the Process:
- Continue this process for each coefficient:
- Multiply the result from the previous step by [tex]\(-10\)[/tex].
- Add this product to the next coefficient in the sequence.
- Fill in the Result Row:
- Perform these steps until you've processed all coefficients.
4. Obtain the Quotient and Remainder:
- The numbers in the result row except for the last number are the coefficients of the quotient polynomial.
- The last number is the remainder.
### Final Result:
After performing synthetic division, the quotient is the polynomial:
[tex]\[ x^3 + 6x^2 + 7x - 7 \][/tex]
The remainder is:
[tex]\[ 0 \][/tex]
Thus, the division of [tex]\( x^4 + 16x^3 + 67x^2 + 63x - 70 \)[/tex] by [tex]\( x + 10 \)[/tex] results in:
[tex]\[ x^3 + 6x^2 + 7x - 7 \][/tex] with a remainder of 0.
This zero remainder confirms that [tex]\( x+10 \)[/tex] is a factor of the given polynomial.
