Let's work through the questions in the image step by step:

Question 3:

In what circumstances might you want to expand brackets? In what circumstances is the factorised form better?

Expanding Brackets:

Expanding brackets is useful in the following situations:

• Simplifying expressions: Expanding helps combine like terms or eliminate brackets in algebraic expressions.
• Solving equations: Expanding can be used when solving equations, especially when combining terms on both sides of the equation.
• Differentiation and Integration: In calculus, expanded forms are easier to differentiate and integrate.
• Substitution: When you need to substitute a value for a variable, having an expanded expression can make the calculation simpler.
• Series approximation: Expanded polynomials are useful for approximations, such as Taylor series.

Factorised Form:

The factorised form is preferable when:

• Solving quadratic equations: Factorisation makes it easier to find the roots or solutions of quadratic equations.
• Identifying roots: A factorised expression shows the roots or zero points of a function clearly.
• Simplifying complex fractions: Factorising allows you to cancel common factors in the numerator and denominator.
• Sketching graphs: Factorised polynomials show intercepts with the axes, helping to sketch curves.
• Efficiency in solving higher-degree polynomials: Factorisation simplifies the solving process for polynomials of higher degrees.

Question 4:

a. Is it always true that the sum of a polynomial of degree $n$ and a polynomial of degree $n-1$ has degree $n$?

Yes, it is true.

• The degree of the sum of two polynomials is determined by the highest degree term among them.
• In this case, the polynomial of degree $n$ will dominate because it has the highest power.
• Adding a polynomial of degree $n-1$ will not affect the highest degree term, so the result still has degree $n$.

b. Is it always true that the sum of a polynomial of degree $n$ and a polynomial of degree $n$ has degree $n$?

Not necessarily.

• If the leading terms of the two polynomials of degree $n$ cancel each other out (i.e., if their coefficients sum to zero), the degree of the resulting polynomial will drop.
• In such a case, the degree could be less than $n$, depending on the remaining terms. If the leading terms don't cancel, the result will have degree $n$.

Would you like further clarification on any part of the solution? Here are some related questions you might find helpful:

1. What is the general method for expanding brackets?
2. How do you factorise quadratic expressions efficiently?
3. Can you explain the difference between a leading coefficient and the degree of a polynomial?
4. How do you determine the roots of a polynomial from its factorised form?
5. What are some methods for solving polynomials of higher degrees?

Tip: When solving equations involving polynomials, always check the highest degree term carefully—it often determines the overall behavior of the function.