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Chapter 0: Problem 45

Factor the difference of two squares. $$x^{4}-16$$

Short Answer

Expert verified
The factored form of the expression \(x^{4}-16\) is \((x^2 + 4)(x+2)(x-2)\)

Step by step solution

01

- Identify The Difference of Squares

The given expression is \(x^{4}-16\), which is a difference of two squares since it can be written as \((x^2)^2 - (4)^2\). Hence, 'a' is \(x^2\) and 'b' is 4 in the formula
02

- Apply the formula for Factoring a Difference of Squares

For the equation, \(a^2 - b^2 = (a+b)(a-b)\), substitute \(x^2\) for 'a' and 4 for 'b'. This gives \((x^2 + 4)(x^2 - 4)\)
03

- Factor further if possible

Notice that the second part of the expression \((x^2 - 4)\) is itself a difference of squares and thus can be factored once again using the formula. Hence \((x^2 - 4)\) can be written as \((x+2)(x-2)\), leading to the fully factored form of the expression as \((x^2 + 4)(x+2)(x-2)\)

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Most popular questions from this chapter

Explain how to multiply rational expressions.

Exercises \(142-144\) will help you prepare for the material covered in the next section. $$\text { Multiply: }\left(2 x^{3} y^{2}\right)\left(5 x^{4} y^{7}\right)$$

$$\text { Solve for } t: \quad s=-16 t^{2}+v_{0} t$$

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.

This will help you prepare for the material covered in the next section. Evaluate $$\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}$$ for \(a=2, b=9,\) and \(c=-5\)
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