Answer :
To fill in the blanks for the division [tex]\( x + 5 \longdiv { x^3 + 2x + 5 } \)[/tex] using synthetic division, follow these steps:
1. Identify the coefficients: The polynomial [tex]\( x^3 + 2x + 5 \)[/tex] can be expressed with all terms by adding a missing term: [tex]\( x^3 + 0x^2 + 2x + 5 \)[/tex]. So, the coefficients you will be working with are: 1 for [tex]\( x^3 \)[/tex], 0 for [tex]\( x^2 \)[/tex], 2 for [tex]\( x \)[/tex], and 5 as the constant term.
2. Determine the synthetic division number: The divisor is [tex]\( x + 5 \)[/tex]. To perform synthetic division, you take the opposite sign of the constant term in the divisor. So, instead of 5, use [tex]\(-5\)[/tex].
3. Set up synthetic division: Write the coefficients [1, 0, 2, 5] horizontally. First, bring down the leading coefficient, which is 1.
4. Perform Synthetic Division:
- Multiply [tex]\(-5\)[/tex] by the last number you wrote (initially, this is the leading coefficient 1) and add the result to the next coefficient (0). This gives you: [tex]\(-5 \times 1 + 0 = -5\)[/tex].
- Continue this process: Multiply [tex]\(-5\)[/tex] by the current result and add to the next coefficient (2). This gives you: [tex]\(-5 \times -5 + 2 = 22\)[/tex].
5. Fill in the blanks:
- The first blank is filled with [tex]\(-4\)[/tex].
- The second blank is filled with [tex]\(22\)[/tex].
Therefore, the blanks in the sequence are:
[tex]\[ -4, \quad 1, \quad 25 \][/tex]
1. Identify the coefficients: The polynomial [tex]\( x^3 + 2x + 5 \)[/tex] can be expressed with all terms by adding a missing term: [tex]\( x^3 + 0x^2 + 2x + 5 \)[/tex]. So, the coefficients you will be working with are: 1 for [tex]\( x^3 \)[/tex], 0 for [tex]\( x^2 \)[/tex], 2 for [tex]\( x \)[/tex], and 5 as the constant term.
2. Determine the synthetic division number: The divisor is [tex]\( x + 5 \)[/tex]. To perform synthetic division, you take the opposite sign of the constant term in the divisor. So, instead of 5, use [tex]\(-5\)[/tex].
3. Set up synthetic division: Write the coefficients [1, 0, 2, 5] horizontally. First, bring down the leading coefficient, which is 1.
4. Perform Synthetic Division:
- Multiply [tex]\(-5\)[/tex] by the last number you wrote (initially, this is the leading coefficient 1) and add the result to the next coefficient (0). This gives you: [tex]\(-5 \times 1 + 0 = -5\)[/tex].
- Continue this process: Multiply [tex]\(-5\)[/tex] by the current result and add to the next coefficient (2). This gives you: [tex]\(-5 \times -5 + 2 = 22\)[/tex].
5. Fill in the blanks:
- The first blank is filled with [tex]\(-4\)[/tex].
- The second blank is filled with [tex]\(22\)[/tex].
Therefore, the blanks in the sequence are:
[tex]\[ -4, \quad 1, \quad 25 \][/tex]
