Question

What is the degree of the polynomial $(x + 3)(x - 8)$?

Hint:

Degree rules:
Product of two linear polynomials → degree 2
Add degrees when multiplying polynomials
Example: $(x)(x)$ gives degree $2$.

Answer 1

Step 1: Understanding the Question:
We are asked to determine the degree of a polynomial that is given in its factored form, \((x + 3)(x - 8)\).
Step 2: Key Formula or Approach:
The degree of a polynomial is the highest power (exponent) of the variable in the polynomial after it has been fully expanded and simplified. Therefore, the first step is to multiply the factors to get the standard form of the polynomial.
Step 3: Detailed Explanation:
The given polynomial is \((x + 3)(x - 8)\).
To expand this product, we can use the distributive property (or the FOIL method):
\[ (x + 3)(x - 8) = x(x - 8) + 3(x - 8) \] Distribute the terms:
\[ = (x \cdot x - x \cdot 8) + (3 \cdot x - 3 \cdot 8) \] \[ = x^2 - 8x + 3x - 24 \] Combine the like terms (the terms containing \(x\)):
\[ = x^2 - 5x - 24 \] The expanded form of the polynomial is \(x^2 - 5x - 24\). The terms are \(x^2\), \(-5x^1\), and \(-24\). The highest power of the variable \(x\) is 2.
Step 4: Final Answer:
The degree of the polynomial \((x + 3)(x - 8)\) is 2.
\[ \boxed{2} \]