Question

If a polynomial is divided by (x + 3), the quotient is x² - x + 7, and the remainder is -4. Find the original polynomial. The original polynomial is [Please provide the expression for the original polynomial].

Answer 1

To find the original polynomial, multiply the divisor (x + 3) by the quotient (x² - x + 7) and add the remainder (-4). Simplifying, the original polynomial is x³ + 2x² + 4x + 17.

Step-by-step explanation:

If a polynomial is divided by (x + 3), and the quotient is x² - x + 7, with a remainder of -4, we can use polynomial long division to reconstruct the original polynomial. According to the remainder theorem, the relationship between the dividend (the original polynomial), the divisor, the quotient, and the remainder is given by:

Dividend = (Divisor × Quotient) + Remainder

In this case, substituting the given values gives us:

Original Polynomial = (x + 3) × (x² - x + 7) - 4

Expanding this we get:

Original Polynomial = x³ - x² + 7x + 3x² - 3x + 21 - 4

Simplify by combining like terms:

Original Polynomial = x³ + 2x² + 4x + 17