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Chapter 2: Problem 49

Verify that \(x^{4}+1=\left(x^{2}+\sqrt{2} x+1\right)\left(x^{2}-\sqrt{2} x+1\right)\).

Short Answer

Expert verified
To verify that \(x^4 + 1 = (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1)\), we multiply the expressions on the right side and simplify. After combining like terms, we get the final expression \(x^4 + 1\), which is equal to the expression on the left side. Therefore, the given expressions are verified: \(x^4 + 1 = (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1)\).

Step by step solution

01

Identify the given expressions

We are given the following expressions to verify: Left side expression (LHS): \(x^4 + 1\) Right side expression (RHS): \((x^2 + \sqrt{2}x + 1) (x^2 - \sqrt{2}x + 1)\)
02

Distribute the terms of the first expression with the terms of the second expression

Multiply the terms in the first expression with the terms in the second expression: \((x^2 + \sqrt{2}x + 1) (x^2 - \sqrt{2}x + 1)\)
03

Apply the distributive property to the expressions

Apply the distributive property and simplify the expressions: \(x^2(x^2 - \sqrt{2}x + 1) + \sqrt{2}x(x^2 - \sqrt{2}x + 1) + 1(x^2 - \sqrt{2}x + 1)\)
04

Multiply and add the terms

Now, multiply the terms and add them to simplify the expression: \(x^4 - \sqrt{2}x^3 + x^2 + \sqrt{2}x^3 - 2x^2 + \sqrt{2}x + x^2 - \sqrt{2}x + 1\)
05

Combine like terms

Now, combine the like terms in the expression: \(x^4 + (-\sqrt{2}x^3 + \sqrt{2}x^3) + (x^2 - 2x^2 + x^2) + (-\sqrt{2}x + \sqrt{2}x) + 1\) \((-\sqrt{2}x^3 + \sqrt{2}x^3)\) and \((-2x^2 + x^2)\) and \((-\sqrt{2}x + \sqrt{2}x)\) are all equal to 0, so we can simplify the expression as: \(x^4 + 1\)
06

Compare the results

We can now see that after multiplying and simplifying the expressions, the right side expression is equal to the left side expression: \(x^4 + 1 = x^4 +1\) So the given expressions are verified: \(x^4 + 1 = (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1)\)

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Most popular questions from this chapter

Verify that \(x^{3}+y^{3}=(x+y)\left(x^{2}-x y+y^{2}\right)\).

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{4 x-5}{x+7} $$

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Write the indicated expression as \(a\) polynomial. $$ \frac{s(1+x)-s(1)}{x} $$

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