Polynomial Expansion 18F47B

1. **State the problem:** We need to expand the cubic polynomial $f(x) = x^3 - [blank]x^2 + [blank]x - [blank]$ by filling in the blanks with appropriate coefficients. 2. **General form of a cubic polynomial:** A cubic polynomial can be written as $$f(x) = ax^3 + bx^2 + cx + d$$ where $a$, $b$, $c$, and $d$ are constants. 3. **Understanding the blanks:** The blanks represent unknown coefficients for the terms $x^2$, $x$, and the constant term. 4. **If the polynomial is given as a product of factors, for example:** $$f(x) = (x - r_1)(x - r_2)(x - r_3)$$ we can expand it step-by-step: 5. **Step 1: Expand the first two factors:** $$ (x - r_1)(x - r_2) = x^2 - (r_1 + r_2)x + r_1 r_2 $$ 6. **Step 2: Multiply the result by the third factor:** $$ (x^2 - (r_1 + r_2)x + r_1 r_2)(x - r_3) $$ 7. **Step 3: Distribute:** $$ x^3 - r_3 x^2 - (r_1 + r_2) x^2 + (r_1 + r_2) r_3 x + r_1 r_2 x - r_1 r_2 r_3 $$ 8. **Step 4: Combine like terms:** $$ x^3 - (r_1 + r_2 + r_3) x^2 + (r_1 r_2 + r_2 r_3 + r_1 r_3) x - r_1 r_2 r_3 $$ 9. **Final expanded form:** $$ f(x) = x^3 - (r_1 + r_2 + r_3) x^2 + (r_1 r_2 + r_2 r_3 + r_1 r_3) x - r_1 r_2 r_3 $$ This shows the coefficients for the blanks in terms of the roots $r_1$, $r_2$, and $r_3$. **Summary:** - The coefficient of $x^2$ is the negative sum of the roots. - The coefficient of $x$ is the sum of the products of the roots taken two at a time. - The constant term is the negative product of all three roots. If you provide specific roots or factors, I can compute the exact coefficients for the blanks.