Evaluating Algebraic Expressions And Polynomials With Radicals

In this section, we will delve into the evaluation of the expression $(\sqrt{5}+\sqrt{2})^2+(\sqrt{8}-\sqrt{5})^2$. This problem involves the manipulation of square roots and the application of algebraic identities. To solve this, we need to carefully expand the squared terms and simplify the resulting expression. Let's break down the steps involved in solving this mathematical expression.

First, we'll tackle the expansion of each squared term separately. The first term, $(\sqrt{5}+\sqrt{2})^2$, can be expanded using the formula $(a+b)^2 = a^2 + 2ab + b^2$. Here, $a = \sqrt{5}$ and $b = \sqrt{2}$. Applying the formula, we get:

$(\sqrt{5}+\sqrt{2})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{2}) + (\sqrt{2})^2$

Simplifying this gives us:

$5 + 2\sqrt{10} + 2$

Which further simplifies to:

$7 + 2\sqrt{10}$

Next, let's expand the second term, $(\sqrt{8}-\sqrt{5})^2$. We can use the formula $(a-b)^2 = a^2 - 2ab + b^2$, where $a = \sqrt{8}$ and $b = \sqrt{5}$. Applying the formula, we have:

$(\sqrt{8}-\sqrt{5})^2 = (\sqrt{8})^2 - 2(\sqrt{8})(\sqrt{5}) + (\sqrt{5})^2$

Simplifying, we get:

$8 - 2\sqrt{40} + 5$

Since $\sqrt{40}$ can be written as $\sqrt{4 \times 10} = 2\sqrt{10}$, the expression becomes:

$13 - 4\sqrt{10}$

Now, we add the simplified forms of both expanded terms:

$(7 + 2\sqrt{10}) + (13 - 4\sqrt{10})$

Combining like terms, we get:

$7 + 13 + 2\sqrt{10} - 4\sqrt{10}$

This simplifies to:

$20 - 2\sqrt{10}$

Therefore, the final evaluation of the expression $(\sqrt{5}+\sqrt{2})^2+(\sqrt{8}-\sqrt{5})^2$ is $20 - 2\sqrt{10}$. This result highlights the importance of understanding algebraic identities and the properties of square roots in simplifying complex expressions. The step-by-step approach ensures accuracy and clarity in the solution.

Now, let's tackle the second part of the problem, which involves finding the value of the expression $x^3 - 3x^2 - 5x + 3$ given that $x = 15\sqrt{2}$. This requires substituting the value of $x$ into the polynomial and simplifying. This problem tests our ability to work with algebraic expressions and substitute values correctly. The solution involves careful substitution and simplification to arrive at the final answer. This section will meticulously walk through the substitution and simplification process, ensuring a clear understanding of each step involved.

We are given that $x = 15\sqrt{2}$. Our goal is to find the value of the expression $x^3 - 3x^2 - 5x + 3$. The first step is to substitute the given value of $x$ into the expression. This yields:

$(15\sqrt{2})^3 - 3(15\sqrt{2})^2 - 5(15\sqrt{2}) + 3$

Now, let's simplify each term individually. First, we'll simplify $(15\sqrt{2})^3$. This can be written as:

$(15\sqrt{2})^3 = 15^3 \times (\sqrt{2})^3 = 3375 \times 2\sqrt{2} = 6750\sqrt{2}$

Next, we simplify $3(15\sqrt{2})^2$. This is:

$3(15\sqrt{2})^2 = 3 \times 15^2 \times (\sqrt{2})^2 = 3 \times 225 \times 2 = 1350$

Then, we simplify $5(15\sqrt{2})$:

$5(15\sqrt{2}) = 75\sqrt{2}$

Now, we substitute these simplified terms back into the original expression:

$6750\sqrt{2} - 1350 - 75\sqrt{2} + 3$

Combining like terms, we have:

$(6750\sqrt{2} - 75\sqrt{2}) + (-1350 + 3)$

This simplifies to:

$6675\sqrt{2} - 1347$

Therefore, the value of the expression $x^3 - 3x^2 - 5x + 3$ when $x = 15\sqrt{2}$ is $6675\sqrt{2} - 1347$. This solution demonstrates the importance of careful substitution and simplification in algebraic problems. By breaking down the problem into smaller parts and simplifying each term individually, we can arrive at the correct answer. The final result underscores the necessity of precision in mathematical calculations.

In summary, we have successfully evaluated the expression $(\sqrt{5}+\sqrt{2})^2+(\sqrt{8}-\sqrt{5})^2$ and found it to be $20 - 2\sqrt{10}$. Additionally, we determined the value of $x^3 - 3x^2 - 5x + 3$ when $x = 15\sqrt{2}$ to be $6675\sqrt{2} - 1347$. These problems showcase the significance of mastering algebraic identities, square root manipulations, and the process of substitution in solving mathematical expressions and equations. The step-by-step solutions provided offer a clear understanding of the methodologies employed, highlighting the precision and accuracy required in mathematical calculations. The ability to tackle such problems is fundamental in advanced mathematics and various scientific disciplines, making the comprehension of these techniques invaluable.