Let's solve the exercises step-by-step. I'll start with Exercise I and work through each part.

Exercise I

Given the expressions: $A = 2x^3 + 3x^2 - 8x + 3$ $B = (x + 3)(x + 2) + x^2 - 9$

1. Show that $A = (x - 1)(2x^2 + 5x - 3)$

We need to factorize $A$ and verify that it can be written as $(x - 1)(2x^2 + 5x - 3)$.

Let's proceed by dividing $A$ by $x - 1$ using polynomial division or synthetic division and check if the result is $2x^2 + 5x - 3$.

2. Expand and simplify $B$

We expand $B = (x + 3)(x + 2) + x^2 - 9$ and simplify the expression.

3. Factorize $B$ and deduce a factorization of $A$

After simplifying $B$, we attempt to factorize it and check if it has a common factor with $A$ to deduce a factorization of $A$.

Exercise II

Part I

Prove the expression: $\frac{3}{5} (ab)^3 \cdot b^3 \cdot x^{a^5} = \frac{1}{5} b^x \cdot a^4$ This requires manipulating exponents and ensuring both sides of the equation are equivalent by using properties of powers.

Part II

Convert the following to scientific notation:

1. $A = 17 \times 10^4 \times 50000 \times 10^{-6}$
2. $B = \frac{3 \times 10^5 \times 7.2 \times 10^{-7}}{2.5 \times 10^{-8}}$

Exercise III

Part I

Simplify and calculate:

1. $A = 5\sqrt{12} + 3\sqrt{3} + \sqrt{75}$
2. $B = \sqrt{96} + 2\sqrt{6} - 2\sqrt{24} - 3\sqrt{54}$

Simplify each square root and combine like terms.

Part II

Rationalize the denominators:

1. $A = \frac{2\sqrt{2} + 3}{2\sqrt{2} + \sqrt{3}}$
2. $B = \frac{2}{1 - \sqrt{2} + \sqrt{3}}$
3. $C = \frac{3\sqrt{3} - 2}{\sqrt{3} - 1} + \frac{\sqrt{3} + 1}{\sqrt{3} + \sqrt{2}}$

To rationalize these, multiply the numerator and denominator by the conjugate of the denominator where necessary.

Would you like me to start with a specific problem or provide solutions in detail for each part?

Related Questions

1. How do we apply synthetic division for polynomial factorization?
2. What are the properties of exponents used in simplifying expressions like in Exercise II?
3. How is scientific notation helpful in simplifying large or small numbers?
4. What are the common methods for simplifying radicals?
5. How does rationalizing the denominator affect the value of an expression?

Tip

For expressions with radicals in the denominator, use conjugates to eliminate the radicals efficiently.