To solve the problem of dividing the polynomial [tex]\(6x^2 - 7x + 5\)[/tex] by [tex]\(2x + 1\)[/tex], we will perform polynomial division. Here is the step-by-step process:
1. Set up the Division:
- Dividend: [tex]\(6x^2 - 7x + 5\)[/tex]
- Divisor: [tex]\(2x + 1\)[/tex]
2. Perform the Division:
- First Term of the Quotient:
- Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{6x^2}{2x} = 3x
\][/tex]
- Multiply the entire divisor by [tex]\(3x\)[/tex]:
[tex]\[
(2x + 1) \times 3x = 6x^2 + 3x
\][/tex]
- Subtract this product from the original polynomial:
[tex]\[
(6x^2 - 7x + 5) - (6x^2 + 3x) = -10x + 5
\][/tex]
- Second Term of the Quotient:
- Divide the new leading term of the remaining polynomial by the leading term of the divisor:
[tex]\[
\frac{-10x}{2x} = -5
\][/tex]
- Multiply the entire divisor by [tex]\(-5\)[/tex]:
[tex]\[
(2x + 1) \times -5 = -10x - 5
\][/tex]
- Subtract this product from the remaining polynomial:
[tex]\[
(-10x + 5) - (-10x - 5) = 10
\][/tex]
3. Conclude the Division:
- The quotient is obtained from the terms found during the division process: [tex]\(3x - 5\)[/tex]
- The remainder is the constant left after the subtraction: [tex]\(10\)[/tex]
Therefore, the quotient and remainder when [tex]\(6x^2 - 7x + 5\)[/tex] is divided by [tex]\(2x + 1\)[/tex] are:
- Quotient: [tex]\(3x - 5\)[/tex]
- Remainder: [tex]\(10\)[/tex]
Thus, we have:
[tex]\[
\left(6x^2 - 7x + 5\right) \div \left(2x + 1\right) = 3x - 5 \quad \text{with a remainder of} \quad 10
\][/tex]
