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Chapter 6: Problem 114

Explain how to factor \(x^{3}+1\)

Short Answer

Expert verified
The factored form of \(x^{3} + 1\) is \((x + 1)(x^{2} - x + 1)\).

Step by step solution

01

Identify the Values for 'a' and 'b'

Examine the equation \(x^{3} + 1\). Here, 'a' is represented by 'x' and 'b' is represented by '1'. So, 'a' = x and 'b' = 1.
02

Plug the Values into the Formula

Now, plug these values into the sum of cubes formula. \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). So, we get \(x^{3} + 1 = (x+1)(x^{2} - x*1 + 1^{2})\).
03

Simplify the Result

The last step is to simplify the result. We obtain: \(x^{3} + 1 = (x+1)(x^{2} - x + 1)\). Then, expand the brackets to verify the result.

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Most popular questions from this chapter

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \((x+3)(3 x+5)=7\)

Use the \(x\)-intercepts for the graph in \(a[-10,10,1]\) by \([-13,10,1]\) viewing rectangle to solve the quadratic equation. Check by substitution. Use the graph of \(y=x^{2}+x-6\) to solve \(x^{2}+x-6=0\).

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &3 x^{3}-12 x^{2}-15 x=3 x(x+5)(x-1) ;[-5,7,1] \text { by }\\\ &[-80,80,10] \end{aligned}$$

Exercises \(150-152\) will help you prepare for the material covered in the next section. Factor: \((x-2)(x+3)-6\)

In Exercises \(130-133,\) use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{3}-1=(x-1)\left(x^{2}-x+1\right)$$
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