Answer :
To divide the polynomial [tex]\(5x^4 + 2x^3 + x^2 + 5\)[/tex] by [tex]\(x - 2\)[/tex], we can use synthetic division. Here is how you can do it step-by-step:
### Step 1: Set up the Synthetic Division
1. Write down the coefficients of the polynomial: [tex]\(5, 2, 1, 0, 5\)[/tex]. Notice that we include a [tex]\(0\)[/tex] for the [tex]\(x\)[/tex] term since it is missing.
2. The divisor is [tex]\(x - 2\)[/tex], which means we use [tex]\(2\)[/tex] for the synthetic division (the root of [tex]\(x - 2\)[/tex]).
### Step 2: Perform the Synthetic Division
- First Row (Bring down the leading coefficient):
- Bring down the first coefficient, [tex]\(5\)[/tex].
- Subsequent Rows:
- Multiply [tex]\(5\)[/tex] (the number you just brought down) by [tex]\(2\)[/tex] and place the result under the next coefficient, which is [tex]\(2\)[/tex].
- Add down the column: [tex]\(2 + (5 \times 2) = 2 + 10 = 12\)[/tex].
- Continue this process:
- Multiply [tex]\(12\)[/tex] by [tex]\(2\)[/tex], and place the result under the next coefficient, [tex]\(1\)[/tex].
- Add down the column: [tex]\(1 + (12 \times 2) = 1 + 24 = 25\)[/tex].
- Multiply [tex]\(25\)[/tex] by [tex]\(2\)[/tex], and place the result under the next coefficient, [tex]\(0\)[/tex].
- Add down the column: [tex]\(0 + (25 \times 2) = 0 + 50 = 50\)[/tex].
- Multiply [tex]\(50\)[/tex] by [tex]\(2\)[/tex], and place the result under the next coefficient, [tex]\(5\)[/tex].
- Add down the column: [tex]\(5 + (50 \times 2) = 5 + 100 = 105\)[/tex].
### Step 3: Interpret the Results
The numbers you have gotten in the second row (ignoring the remainder) represent the coefficients of the quotient polynomial. The last number (105) is the remainder.
- Therefore, the quotient of the division is [tex]\(5x^3 + 12x^2 + 25x + 50\)[/tex].
- The remainder is [tex]\(105\)[/tex].
### Conclusion
The result of dividing [tex]\(5x^4 + 2x^3 + x^2 + 5\)[/tex] by [tex]\(x - 2\)[/tex] is:
[tex]\[
5x^3 + 12x^2 + 25x + 50 \quad \text{with a remainder of} \quad 105
\][/tex]
### Step 1: Set up the Synthetic Division
1. Write down the coefficients of the polynomial: [tex]\(5, 2, 1, 0, 5\)[/tex]. Notice that we include a [tex]\(0\)[/tex] for the [tex]\(x\)[/tex] term since it is missing.
2. The divisor is [tex]\(x - 2\)[/tex], which means we use [tex]\(2\)[/tex] for the synthetic division (the root of [tex]\(x - 2\)[/tex]).
### Step 2: Perform the Synthetic Division
- First Row (Bring down the leading coefficient):
- Bring down the first coefficient, [tex]\(5\)[/tex].
- Subsequent Rows:
- Multiply [tex]\(5\)[/tex] (the number you just brought down) by [tex]\(2\)[/tex] and place the result under the next coefficient, which is [tex]\(2\)[/tex].
- Add down the column: [tex]\(2 + (5 \times 2) = 2 + 10 = 12\)[/tex].
- Continue this process:
- Multiply [tex]\(12\)[/tex] by [tex]\(2\)[/tex], and place the result under the next coefficient, [tex]\(1\)[/tex].
- Add down the column: [tex]\(1 + (12 \times 2) = 1 + 24 = 25\)[/tex].
- Multiply [tex]\(25\)[/tex] by [tex]\(2\)[/tex], and place the result under the next coefficient, [tex]\(0\)[/tex].
- Add down the column: [tex]\(0 + (25 \times 2) = 0 + 50 = 50\)[/tex].
- Multiply [tex]\(50\)[/tex] by [tex]\(2\)[/tex], and place the result under the next coefficient, [tex]\(5\)[/tex].
- Add down the column: [tex]\(5 + (50 \times 2) = 5 + 100 = 105\)[/tex].
### Step 3: Interpret the Results
The numbers you have gotten in the second row (ignoring the remainder) represent the coefficients of the quotient polynomial. The last number (105) is the remainder.
- Therefore, the quotient of the division is [tex]\(5x^3 + 12x^2 + 25x + 50\)[/tex].
- The remainder is [tex]\(105\)[/tex].
### Conclusion
The result of dividing [tex]\(5x^4 + 2x^3 + x^2 + 5\)[/tex] by [tex]\(x - 2\)[/tex] is:
[tex]\[
5x^3 + 12x^2 + 25x + 50 \quad \text{with a remainder of} \quad 105
\][/tex]
