Polynomial Operations 6F829C

1. **Problem:** Find the product $f(n) \cdot g(n)$ where $f(n) = -3n^2 - 2$ and $g(n) = 4n - 5$. 2. **Formula:** To multiply two polynomials, use the distributive property: $(a+b)(c+d) = ac + ad + bc + bd$. 3. **Calculation:** $$f(n) \cdot g(n) = (-3n^2 - 2)(4n - 5)$$ $$= (-3n^2)(4n) + (-3n^2)(-5) + (-2)(4n) + (-2)(-5)$$ $$= -12n^3 + 15n^2 - 8n + 10$$ 4. **Answer:** The product is $-12n^3 + 15n^2 - 8n + 10$. --- 1. **Problem:** Determine which ordered pair satisfies $f(x) = 3x^2 - 1$. 2. **Method:** Substitute the $x$ value from each pair into $f(x)$ and check if the result equals the $y$ value. 3. **Check pairs:** - For $(5, 19)$: $f(5) = 3(5)^2 - 1 = 3(25) - 1 = 75 - 1 = 74 \neq 19$ - For $(1, 13)$: $f(1) = 3(1)^2 - 1 = 3 - 1 = 2 \neq 13$ - For $(0, 5)$: $f(0) = 3(0)^2 - 1 = 0 - 1 = -1 \neq 5$ - For $(-2, 11)$: $f(-2) = 3(-2)^2 - 1 = 3(4) - 1 = 12 - 1 = 11$ which matches the $y$ value. 4. **Answer:** The solution is $(-2, 11)$. --- 1. **Problem:** Find $g(n) - h(n)$ where $g(n) = 4n + 5$ and $h(n) = -2n^2 + 5$. 2. **Formula:** Subtract polynomials by subtracting corresponding terms. 3. **Calculation:** $$g(n) - h(n) = (4n + 5) - (-2n^2 + 5)$$ $$= 4n + 5 + 2n^2 - 5$$ $$= 2n^2 + 4n + \cancel{5} - \cancel{5}$$ $$= 2n^2 + 4n$$ 4. **Answer:** The result is $2n^2 + 4n$.