Sure, let's divide the polynomial [tex]\( x^3 - 3x^2 + 0x + 5 \)[/tex] by [tex]\( x - 3 \)[/tex] using synthetic division. Here is the step-by-step process:
1. Identify the coefficients of the dividend polynomial:
[tex]\[ x^3 - 3x^2 + 0x + 5 \][/tex]
The coefficients are [tex]\( [1, -3, 0, 5] \)[/tex].
2. Identify the root of the divisor [tex]\( x - 3 \)[/tex]:
The root is [tex]\( 3 \)[/tex].
3. Set up for synthetic division:
Write down the coefficients in a row:
[tex]\[ 1 \quad -3 \quad 0 \quad 5 \][/tex]
Draw a synthetic division bar and write the root [tex]\( 3 \)[/tex] to the left of it.
4. Synthetic division process:
- First step: Bring down the leading coefficient:
[tex]\[ \quad 1 \][/tex]
[tex]\[ 3 \, \big| \, 1 \quad -3 \quad 0 \quad 5 \][/tex]
- Multiply this leading coefficient by the root [tex]\( 3 \)[/tex] and write the result beneath the next coefficient:
[tex]\[ \quad 1 \][/tex]
[tex]\[ 3 \, \big| \, 1 \quad -3 \quad 0 \quad 5 \][/tex]
[tex]\[ \quad \quad 3 \][/tex]
- Add the next column:
[tex]\[ \quad 1 \][/tex]
[tex]\[ 3 \, \big| \, 1 \quad 0 \quad 0 \quad 5 \][/tex]
[tex]\[ \quad \quad 3 \][/tex]
- Multiply the result by the root [tex]\( 3 \)[/tex] and write it beneath the next coefficient:
[tex]\[ \quad 1 \][/tex]
[tex]\[ 3 \, \big| \, 1 \quad 0 \quad 0 \quad 5 \][/tex]
[tex]\[ \quad \quad 3 \quad 0 \][/tex]
- Add the next column:
[tex]\[ \quad 1 \][/tex]
[tex]\[ 3 \, \big| \, 1 \quad 0 \quad 0 \quad 5 \][/tex]
[tex]\[ \quad \quad 3 \quad 0 \quad 0 \][/tex]
- Multiply the result by the root [tex]\( 3 \)[/tex] and write it beneath the next coefficient:
[tex]\[ \quad 1 \][/tex]
[tex]\[ 3 \, \big| \, 1 \quad 0 \quad 0 \quad 5 \][/tex]
[tex]\[ \quad \quad 3 \quad 0 \quad 0 \][/tex]
[tex]\[ \quad \quad \quad \quad 0 \][/tex]
- Add the final column to find the remainder:
[tex]\[ \quad 1 \][/tex]
[tex]\[ 3 \, \big| \, 1 \quad 0 \quad 0 \quad 5 \][/tex]
[tex]\[ \quad \quad 3 \quad 0 \quad 0 \][/tex]
[tex]\[ \quad \quad \quad \quad 0 \quad 5 \][/tex]
5. Read the quotient and the remainder:
The quotient is formed by the numbers obtained above the line after performing synthetic division:
[tex]\[ x^2 + 0x + 0 \rightarrow x^2 \][/tex]
The remainder is the last number on the bottom row:
[tex]\[ \text{Remainder} = 5 \][/tex]
Therefore, the result of dividing [tex]\( x^3 - 3x^2 + 5 \)[/tex] by [tex]\( x - 3 \)[/tex] is:
[tex]\[
\left(x^3 - 3x^2 + 5\right) \div (x - 3) = x^2 + 5 \quad \text{with a remainder of } 5
\][/tex]
Or you can express the division as:
[tex]\[
x^3 - 3x^2 + 5 = (x - 3)(x^2) + 5
\][/tex]
