To solve the polynomial division problem [tex]\[(8x^3 - x^2 + 6x + 7) \div (2x - 1),\][/tex] we should start by dividing the leading term of the dividend by the leading term of the divisor.
Here’s the step-by-step approach:
1. Identify the leading terms:
- The leading term of the dividend (8x^3 - x^2 + 6x + 7) is [tex]\(8x^3\)[/tex].
- The leading term of the divisor (2x - 1) is [tex]\(2x\)[/tex].
3. Divide the leading terms:
- When we divide [tex]\(8x^3\)[/tex] by [tex]\(2x\)[/tex], we focus on the coefficients and the exponents separately.
- The coefficient of [tex]\(8x^3\)[/tex] is 8, and the coefficient of [tex]\(2x\)[/tex] is 2.
- The variable part [tex]\(x^3\)[/tex] divided by [tex]\(x\)[/tex] gives [tex]\(x^{3-1} = x^2\)[/tex].
4. Perform the division:
- Dividing the coefficients: [tex]\(\frac{8}{2} = 4\)[/tex].
- The variable part results in [tex]\(x^2\)[/tex].
So, [tex]\(\frac{8x^3}{2x} = 4x^2\)[/tex].
Therefore, the first step in the division is to divide the leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{8x^3}{2x} = 4x^2. \][/tex]
In simpler terms, the result of this division is [tex]\(4x^2\)[/tex].
