what is the degree of the polynomial f(x)=x^2 (x-3)^a(x+2)^3(x+1)^b where a=4 and b=5?

To determine the degree of the polynomial \( f(x) = x^2 (x - 3)^a (x + 2)^3 (x + 1)^b \) with \( a = 4 \) and \( b = 5 \), we can follow these steps:

1. **Substitute the values of \( a \) and \( b \):**
   \[
   f(x) = x^2 (x - 3)^4 (x + 2)^3 (x + 1)^5
   \]

2. **Identify the degree of each term:**
   - The term \( x^2 \) has a degree of \( 2 \).
   - The term \( (x - 3)^4 \) has a degree of \( 4 \).
   - The term \( (x + 2)^3 \) has a degree of \( 3 \).
   - The term \( (x + 1)^5 \) has a degree of \( 5 \).

3. **Add the degrees:**
   The degree of the polynomial \( f(x) \) is the sum of the degrees of all the terms:
   \[
   2 + 4 + 3 + 5 = 14
   \]

Thus, the degree of the polynomial \( f(x) \) is **14**.

Would you like additional details or have any questions?

Here are 5 related questions to deepen understanding:
1. How does the degree change if the exponents \( a \) and \( b \) are different?
2. What is the leading coefficient of the polynomial if expanded fully?
3. How does the degree of a polynomial affect its behavior as \( x \to \infty \) or \( x \to -\infty \)?
4. Can the degree of a polynomial ever be a non-integer?
5. How does the degree relate to the number of roots a polynomial might have?

**Tip:** The degree of a polynomial is the sum of the exponents of its highest-power terms in each factor.

What is the degree of the polynomial f(x)=x^2 (x-3)^a(x+2)^3(x+1)^b where a=4 and b=5?

Calculate the degree of the polynomial f(x) = x^2 (x - 3)^4 (x + 2)^3 (x + 1)^5, with step-by-step details on summing the exponents for each term. This exercise covers polynomial degrees, suitable for students in Grades 10-12.

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