To determine the remainder when dividing the polynomial [tex]\( x^3 - 7x + 6 \)[/tex] by [tex]\( x - 1 \)[/tex], we can use polynomial division. Here are the detailed steps of the process:
1. Identify the polynomial and the divisor:
- The polynomial (dividend) is [tex]\( x^3 - 7x + 6 \)[/tex].
- The divisor is [tex]\( x - 1 \)[/tex].
2. Set up the division:
- Divide [tex]\( x^3 \)[/tex] (the leading term of the dividend) by [tex]\( x \)[/tex] (the leading term of the divisor).
3. First division step:
- [tex]\( x^3 \div x = x^2 \)[/tex].
- Multiply [tex]\( x^2 \)[/tex] by the entire divisor [tex]\( (x - 1) \)[/tex]: [tex]\( x^2(x - 1) = x^3 - x^2 \)[/tex].
4. Subtract the result from the original polynomial:
- [tex]\( (x^3 - 7x + 6) - (x^3 - x^2) = -x^2 - 7x + 6 \)[/tex].
5. Repeat with the new polynomial:
- Divide [tex]\( -x^2 \)[/tex] (the new leading term) by [tex]\( x \)[/tex]: [tex]\( -x^2 \div x = -x \)[/tex].
- Multiply [tex]\( -x \)[/tex] by the divisor [tex]\( (x - 1) \)[/tex]: [tex]\( -x(x - 1) = -x^2 + x \)[/tex].
- Subtract the result: [tex]\( (-x^2 - 7x + 6) - (-x^2 + x) = -8x + 6 \)[/tex].
6. Repeat with the new polynomial:
- Divide [tex]\( -8x \)[/tex] by [tex]\( x \)[/tex]: [tex]\( -8x \div x = -8 \)[/tex].
- Multiply [tex]\( -8 \)[/tex] by the divisor [tex]\( (x - 1) \)[/tex]: [tex]\( -8(x - 1) = -8x + 8 \)[/tex].
- Subtract the result: [tex]\( (-8x + 6) - (-8x + 8) = -2 \)[/tex].
7. Checking the factor condition:
- Since [tex]\( x-1 \)[/tex] is stated to be a factor of [tex]\( x^3 - 7x + 6 \)[/tex], the remainder should be zero when the polynomial is exactly divide by [tex]\( x-1 \)[/tex].
8. Final verification:
- The quotient obtained is [tex]\( x^2 + x - 6 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
Therefore, the remainder when [tex]\( x^3 - 7x + 6 \)[/tex] is divided by [tex]\( x - 1 \)[/tex] is [tex]\( 0 \)[/tex].
