Problem 192 Factor each polynomial completel... [FREE SOLUTION]

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Chapter 5: Problem 192

Factor each polynomial completely. $$t^{4}-1$$

Short Answer

Expert verified

$t^4 - 1 = (t - 1)(t + 1)(t^2 + 1)$

Step by step solution

Identify the form of the polynomial

Notice that the polynomial is in the form of a difference of squares: $t^4 - 1$.

Apply the difference of squares formula

Recall the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$. Here, let $a = t^2$ and $b = 1$. Applying the formula: $t^4 - 1 = (t^2)^2 - 1^2 = (t^2 - 1)(t^2 + 1)$.

Factor further if possible

Next, check to see if each factor can be factored further. Notice that $t^2 - 1$ is also a difference of squares: $t^2 - 1 = (t - 1)(t + 1)$. However, $t^2 + 1$ cannot be factored over the real numbers. Thus, the completely factored form is: $t^4 - 1 = (t^2 - 1)(t^2 + 1) = (t - 1)(t + 1)(t^2 + 1)$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

difference of squares

The difference of squares is a key concept in polynomial factoring. It occurs when you have two terms that are both perfect squares, separated by a subtraction sign. The general formula for the difference of squares is: \[a^2 - b^2 = (a - b)(a + b).\] To apply this, you need to identify the square roots of each term. For instance, in the polynomial \[t^4 - 1,\] we recognize that \[t^4\] is the square of \[t^2,\] and \[1\] is the square of \[1.\] Using the difference of squares formula, this can be rewritten as: \[t^4 - 1 = (t^2 - 1)(t^2 + 1).\] This simplifies the polynomial and sets up the next steps for further factoring.

polynomial factoring

Polynomial factoring involves breaking down a polynomial into simpler factors or products of polynomials that, when multiplied together, result in the original polynomial. This technique simplifies complex expressions and is essential in solving polynomial equations. In our example \[t^4 - 1,\] after applying the difference of squares, we have \[t^2 - 1\] and \[t^2 + 1.\] We then check each factor to see if it can be simplified further. The factor \[t^2 - 1\] is again a difference of squares: \[t^2 - 1 = (t - 1)(t + 1).\] However, \[t^2 + 1\] cannot be factored over the real numbers because it does not fit any common factoring patterns.

completely factored form

A polynomial is in completely factored form when it is expressed as a product of irreducible factors. These factors cannot be factored further using real numbers. For the polynomial \[t^4 - 1,\] the completely factored form is found by breaking it down step by step. Initially, we applied the difference of squares to get \[t^4 - 1 = (t^2 - 1)(t^2 + 1).\] Then, further factoring \[t^2 - 1\] as another difference of squares gives: \[(t^2 - 1) = (t - 1)(t + 1).\] Thus, the completely factored form of \[t^4 - 1\] is: \[t^4 - 1 = (t - 1)(t + 1)(t^2 + 1).\] At this stage, no further factorization is possible using real numbers, and the polynomial is fully simplified.

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91影视

Factor each polynomial completely. $$t^{4}-1$$

Short Answer

Step by step solution

Identify the form of the polynomial

Apply the difference of squares formula

Factor further if possible

Key Concepts

difference of squares

polynomial factoring

completely factored form

One App. One Place for Learning.

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