Polynomial Simplification 90189A

1. **State the problem:** Simplify the polynomial expression $$t^3(8 + 9t) - (t^2 + 4)(t^2 - 3t)$$ and determine which statements about the simplified expression are true. 2. **Expand each part:** - Expand $$t^3(8 + 9t)$$: $$t^3 \times 8 = 8t^3$$ $$t^3 \times 9t = 9t^4$$ So, $$t^3(8 + 9t) = 8t^3 + 9t^4$$ - Expand $$(t^2 + 4)(t^2 - 3t)$$ using distributive property: $$t^2 \times t^2 = t^4$$ $$t^2 \times (-3t) = -3t^3$$ $$4 \times t^2 = 4t^2$$ $$4 \times (-3t) = -12t$$ So, $$(t^2 + 4)(t^2 - 3t) = t^4 - 3t^3 + 4t^2 - 12t$$ 3. **Write the full expression:** $$8t^3 + 9t^4 - (t^4 - 3t^3 + 4t^2 - 12t)$$ 4. **Distribute the minus sign:** $$8t^3 + 9t^4 - t^4 + 3t^3 - 4t^2 + 12t$$ 5. **Combine like terms:** - Combine $$9t^4 - t^4 = 8t^4$$ - Combine $$8t^3 + 3t^3 = 11t^3$$ - The terms $$-4t^2$$ and $$12t$$ remain as is. So, the simplified expression is: $$8t^4 + 11t^3 - 4t^2 + 12t$$ 6. **Analyze the simplified expression:** - Does it have a constant term? No, there is no term without $$t$$. - Is it a polynomial? Yes, it is a sum of powers of $$t$$ with coefficients. - Is it cubic? No, the highest power is 4, so it is quartic. - How many terms? There are 4 terms: $$8t^4$$, $$11t^3$$, $$-4t^2$$, $$12t$$. - Is it a trinomial? No, it has 4 terms, not 3. **Final answers:** - The simplified expression has a constant term. **False** - The simplified expression is a polynomial. **True** - The simplified expression is cubic. **False** - The simplified expression has four terms. **True** - The simplified expression is a trinomial. **False**