From the image provided, I see that you are working on a polynomial function activity, where you need to analyze given polynomials by filling in details such as their standard form, degree, leading coefficient, and constant term.

Let’s start by analyzing the polynomials from the table:

1. $f(x) = 2 - 11x + 2x^2$

• Standard form: To write the polynomial in standard form, we arrange the terms in descending powers of $x$. So, it becomes: $f(x) = 2x^2 - 11x + 2$
• Degree: The highest power of $x$ is 2, so the degree is 2.
• Leading coefficient: The leading term is $2x^2$, so the leading coefficient is 2.
• Constant term: The constant term is the term without $x$, which is 2.

2. $f(x) = x(x - 5)$

• Standard form: First expand the expression: $f(x) = x(x - 5) = x^2 - 5x$
• Degree: The highest power of $x$ is 2, so the degree is 2.
• Leading coefficient: The leading term is $x^2$, so the leading coefficient is 1.
• Constant term: There is no constant term, so it is 0.

3. $y = x(x + 3)(x^2 - 2)$

• Standard form: Let’s expand this step-by-step: $y = x \cdot (x + 3) \cdot (x^2 - 2)$ First, expand $(x + 3)(x^2 - 2)$: $(x + 3)(x^2 - 2) = x^3 - 2x + 3x^2 - 6$ Now multiply by $x$: $y = x(x^3 - 2x + 3x^2 - 6) = x^4 + 3x^3 - 2x^2 - 6x$
• Degree: The highest power of $x$ is 4, so the degree is 4.
• Leading coefficient: The leading term is $x^4$, so the leading coefficient is 1.
• Constant term: There is no constant term, so it is 0.

4. $y = (x + 4)(x + 1)(x - 2)$

• Standard form: Expand step-by-step: $y = (x + 4)(x + 1)(x - 2)$ First expand $(x + 4)(x + 1)$: $(x + 4)(x + 1) = x^2 + x + 4x + 4 = x^2 + 5x + 4$ Now expand $(x^2 + 5x + 4)(x - 2)$: $(x^2 + 5x + 4)(x - 2) = x^3 - 2x^2 + 5x^2 - 10x + 4x - 8 = x^3 + 3x^2 - 6x - 8$
• Degree: The highest power of $x$ is 3, so the degree is 3.
• Leading coefficient: The leading term is $x^3$, so the leading coefficient is 1.
• Constant term: The constant term is -8.

Would you like further details or clarification on any step?

Here are 5 related questions:

1. What is the significance of the leading coefficient in determining the graph of the polynomial?
2. How does the degree of a polynomial affect its end behavior?
3. Can a polynomial have more than one leading term?
4. How do you identify the roots of a polynomial from its factored form?
5. How does expanding a polynomial change its appearance but not its roots?

Tip: When expanding polynomials, always be cautious with the distribution of each term to avoid mistakes.