Is (a - 1)x^3 + x^{2b - 4} + (a + b)x + a · b = 0 a quadratic equation?

Steps to Solve

1. Identify the degree of the polynomial To classify the polynomial as a quadratic equation, we need to find the highest degree term. A polynomial is quadratic if its highest degree term is 2.

2. Examine each term of the polynomial The polynomial given is: $$(a - 1)x^3 + x^{2b - 4} + (a + b)x + a \cdot b$$ Check the degree of each term:

• The first term is $(a - 1)x^3$, which has a degree of 3.
• The second term is $x^{2b - 4}$. The degree depends on the value of $b$.
• The third term is $(a + b)x$, which has a degree of 1.
• The last term, $a \cdot b$, is constant (degree 0).

3. Determine conditions for quadratic For the polynomial to be quadratic, the highest degree must equal 2. We need to analyze the term $x^{2b - 4}$: To ensure the term $x^{2b - 4}$ does not exceed degree 2: $$2b - 4 \leq 2$$ This simplifies to: $$2b \leq 6$$ So, $$b \leq 3$$

4. Determine if term is quadratic under certain conditions The polynomial can only be quadratic if:

• The term $(a - 1)x^3$ does not contribute, which means $a = 1$ to nullify this term. This gives us $0x^3$.
• Additionally, for $b$ to keep $2b - 4$ at most 2, we already established $b \leq 3$.

5. Final conclusion The polynomial can only be classified as a quadratic equation if both conditions are satisfied:

• $a = 1$
• $b \leq 3$