Let's focus on the fourth part of the first question (Question 6, part iv) that you've asked to understand:

Question 6 (iv):

Add the following polynomials: $\frac{2}{3} x^2 - \frac{1}{3} x + 5, \quad \frac{4}{3} x^2 + \frac{2}{3} x - \frac{4}{3}, \quad \frac{5}{3} x^2 - \frac{4}{3} x + 1$

Step-by-Step Solution:

To add these polynomials, we will combine like terms by adding the coefficients of the corresponding terms:

1. Combine the $x^2$ terms:

   $\left( \frac{2}{3} + \frac{4}{3} + \frac{5}{3} \right) x^2 = \left( \frac{2 + 4 + 5}{3} \right) x^2 = \frac{11}{3} x^2$

2. Combine the $x$ terms:

   $\left( -\frac{1}{3} + \frac{2}{3} - \frac{4}{3} \right) x = \left( \frac{-1 + 2 - 4}{3} \right) x = -\frac{3}{3} x = -x$

3. Combine the constant terms:

   $5 - \frac{4}{3} + 1 = \left( 5 + 1 \right) - \frac{4}{3} = 6 - \frac{4}{3} = \frac{18}{3} - \frac{4}{3} = \frac{14}{3}$

Final Answer:

$\frac{11}{3} x^2 - x + \frac{14}{3}$

This is the result of adding the given polynomials.

Would you like more details, or do you have any questions?

Related Questions:

1. How do you add polynomials with different coefficients?
2. What is the importance of combining like terms in polynomial addition?
3. How do you simplify fractions when combining like terms in polynomials?
4. Can you show me how to subtract these polynomials instead?
5. What are the common mistakes to avoid when adding or subtracting polynomials?

Tip:

When adding or subtracting polynomials, always align like terms (terms with the same variable and exponent) to ensure correct operations.