When 2x²−7x+9 is divided by a polynomial, the quotient is 2x-3 and the remainder is 3. Find the polynomial.

Verified step by step guidance

1. Recall the polynomial division identity: Dividend = (Divisor) × (Quotient) + Remainder.
2. Let the divisor polynomial be denoted as \(D(x)\). According to the problem, the dividend is \$2x^{2} - 7x + 9\(, the quotient is \)2x - 3$, and the remainder is 3.
3. Set up the equation: \$2x^{2} - 7x + 9 = D(x) \times (2x - 3) + 3$.
4. Rearrange the equation to isolate \(D(x)\): \(D(x) = \frac{2x^{2} - 7x + 9 - 3}{2x - 3} = \frac{2x^{2} - 7x + 6}{2x - 3}\).
5. Perform polynomial division or factorization on the numerator \$2x^{2} - 7x + 6\( to simplify the expression and find \)D(x)$.

Key Concepts

Polynomial Division

Polynomial division involves dividing one polynomial by another, resulting in a quotient and a remainder. The dividend equals the divisor multiplied by the quotient plus the remainder. Understanding this relationship is essential to reconstruct the divisor when the dividend, quotient, and remainder are known.

Forming Equations from Polynomial Expressions

Given the dividend, quotient, and remainder, you can set up an equation: Dividend = Divisor × Quotient + Remainder. This equation allows you to solve for the unknown polynomial divisor by expressing it in terms of the known polynomials and constants.

Solving for Unknown Polynomials

To find the unknown polynomial divisor, rearrange the equation and perform algebraic operations such as polynomial subtraction and division. Matching coefficients of corresponding powers of x helps determine the coefficients of the unknown polynomial.