1. **Problem Statement:** Find the number that must be added to the polynomial $$2x^3 - 7x^2 + 2x$$ so that when the resulting polynomial is divided by $$2x - 3$$, the remainder is $$-2$$. 2. **Formula and Important Rule:** By the Remainder Theorem, the remainder when a polynomial $$f(x)$$ is divided by a linear divisor $$ax - b$$ is $$f\left(\frac{b}{a}\right)$$. 3. **Apply the Remainder Theorem:** Here, divisor is $$2x - 3$$, so $$a=2$$ and $$b=3$$. The remainder when dividing by $$2x - 3$$ is $$f\left(\frac{3}{2}\right)$$. 4. **Evaluate the original polynomial at $$x=\frac{3}{2}$$:** $$f\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right)^3 - 7\left(\frac{3}{2}\right)^2 + 2\left(\frac{3}{2}\right)$$ Calculate each term: - $$\left(\frac{3}{2}\right)^3 = \frac{27}{8}$$ - $$\left(\frac{3}{2}\right)^2 = \frac{9}{4}$$ So, $$f\left(\frac{3}{2}\right) = 2 \times \frac{27}{8} - 7 \times \frac{9}{4} + 2 \times \frac{3}{2} = \frac{54}{8} - \frac{63}{4} + 3$$ Simplify fractions: - $$\frac{54}{8} = \frac{27}{4}$$ - $$\frac{63}{4}$$ stays as is So, $$f\left(\frac{3}{2}\right) = \frac{27}{4} - \frac{63}{4} + 3 = -\frac{36}{4} + 3 = -9 + 3 = -6$$ 5. **Let the number to be added be $$k$$.** Then the new polynomial is $$2x^3 - 7x^2 + 2x + k$$. The remainder when divided by $$2x - 3$$ is: $$f\left(\frac{3}{2}\right) + k = -6 + k$$ 6. **Set the remainder equal to $$-2$$ and solve for $$k$$:** $$-6 + k = -2$$ $$k = -2 + 6 = 4$$ **Final answer:** The number to be added is **4**.