The given problem is:

The polynomial $x^3 + 7x^2 + ax + b$ has factors $(x - 2)$ and $(x + 2)$. We are required to find the values of $a$ and $b$, and the third factor.

Step-by-step solution:

Since $(x - 2)$ and $(x + 2)$ are factors of the cubic polynomial, their product is part of the factorization. First, we calculate:

$(x - 2)(x + 2) = x^2 - 4$

Thus, we can express the cubic polynomial as: $x^3 + 7x^2 + ax + b = (x^2 - 4)(x + k)$ where $k$ is the unknown third factor we need to determine.

Now expand the right-hand side:

$(x^2 - 4)(x + k) = x^3 + kx^2 - 4x - 4k$

Comparing this with the original polynomial $x^3 + 7x^2 + ax + b$, we can match the coefficients:

• Coefficient of $x^3$: $1 = 1$ (this is already correct).
• Coefficient of $x^2$: $k = 7$, so $k = 7$.
• Coefficient of $x$: $-4 + 7 = a$, so $a = 3$.
• Constant term: $-4k = b$, so $b = -4 \times 7 = -28$.

Final values:

• $a = 3$
• $b = -28$
• The third factor is $(x + 7)$.

Would you like more details on any step or further explanation? Here are five related questions you might explore:

1. How do you factor higher-degree polynomials with multiple factors?
2. Why does comparing coefficients work when expanding factored forms of polynomials?
3. Can you apply synthetic division to verify the factors of a polynomial?
4. What is the remainder theorem, and how is it related to polynomial factorization?
5. How would you solve for unknown coefficients if given four or more factors?

Tip: When expanding factored polynomials, always compare each term systematically to avoid errors in the coefficients.