When 2x^2−7x+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. Step 1: Recall the division algorithm for polynomials. It states that if a polynomial f(x) is divided by a polynomial g(x), then there exist unique polynomials q(x) and r(x) such that f(x) = g(x)q(x) + r(x), where the degree of r(x) is less than the degree of g(x). In this case, f(x) = 2x^2−7x+9, q(x) = 2x-3, and r(x) = 3.
2. Step 2: We are asked to find g(x). We can rearrange the division algorithm equation to solve for g(x). This gives us g(x) = (f(x) - r(x)) / q(x).
3. Step 3: Substitute the given values into the equation from step 2. This gives us g(x) = ((2x^2−7x+9) - 3) / (2x-3).
4. Step 4: Simplify the numerator of the fraction on the right side of the equation. This gives us g(x) = (2x^2−7x+6) / (2x-3).
5. Step 5: Now, perform the division on the right side of the equation to find g(x).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. The result consists of a quotient and a remainder. In this context, the polynomial being divided is 2x^2−7x+9, and the division yields a quotient of 2x-3 and a remainder of 3.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear polynomial of the form (x - c), the remainder of this division is f(c). In this problem, the remainder is given as 3, which helps in determining the polynomial that divides 2x^2−7x+9.

Constructing Polynomials

Constructing a polynomial involves using the quotient and remainder from polynomial division to express the original polynomial. The relationship can be expressed as f(x) = (divisor)(quotient) + remainder. Here, we can find the divisor by rearranging this equation using the known quotient and remainder.