Let's work through the problem in the image step by step.

Question 2a

We are given the polynomial function:

$f(x) = (x + 1)(x + 3)(x - 4)$

To rewrite this polynomial in standard form:

We'll expand the factored form to get a polynomial of the form $ax^n + bx^{n-1} + \dots$.

Step 1: Multiply the first two factors:

$(x + 1)(x + 3) = x^2 + 3x + x + 3 = x^2 + 4x + 3$

Step 2: Now, multiply the result by the third factor $(x - 4)$:

$(x^2 + 4x + 3)(x - 4)$

Distribute each term of $x^2 + 4x + 3$ to $x - 4$:

$x^2(x - 4) = x^3 - 4x^2$ $4x(x - 4) = 4x^2 - 16x$ $3(x - 4) = 3x - 12$

Step 3: Combine all the terms:

$x^3 - 4x^2 + 4x^2 - 16x + 3x - 12$

Simplify: $f(x) = x^3 - 13x - 12$

Degree and constant term:

• Degree: The highest power of $x$ is 3, so the degree is 3.
• Constant term: The constant term is the term without $x$, which is $-12$.

Final Answer:

$f(x) = x^3 - 13x - 12$

• Degree: 3
• Constant term: $-12$

Do you have any questions or would you like further clarification on this solution?

Here are 5 related questions for further practice:

1. Expand and simplify $f(x) = (x - 2)(x + 5)(x - 1)$.
2. What is the degree and constant term of the polynomial $f(x) = (x - 3)(x + 2)(x + 4)$?
3. How would the graph of $f(x) = (x + 1)(x + 3)(x - 4)$ look based on its roots and degree?
4. Rewrite $f(x) = (x + 1)(x + 2)(x + 3)(x - 4)$ in standard form.
5. What is the behavior of the polynomial $f(x) = x^3 - 13x - 12$ as $x$ approaches positive and negative infinity?

Tip:

When expanding polynomials, distribute carefully and keep track of all terms to avoid mistakes in combining like terms!