Problem 152 In the following exercises, fact... [FREE SOLUTION]

Chapter 6: Problem 152

In the following exercises, factor each expression using any method. $$ x^{4}-7 x^{2}-8 $$

Short Answer

Expert verified

The factored form is $ (x - 2\sqrt{2})(x + 2\sqrt{2})(x^2 + 1) $.

Step by step solution

Identify the quadratic form

Recognize that the given expression, $ x^4 - 7x^2 - 8 $, can be seen as a quadratic in terms of $ x^2 $. Let $ y = x^2 $, which transforms the expression into $ y^2 - 7y - 8 $.

Factor the quadratic expression

Factor the quadratic expression $ y^2 - 7y - 8 $. Look for two numbers that multiply to give $-8$ and add to give $-7$. These numbers are $-8$ and $1$. Thus, the expression factors as $ (y - 8)(y + 1) $.

Substitute back $ x^2 $ for $ y $

Replace $ y $ with $ x^2 $ in the factored form. This gives $ (x^2 - 8)(x^2 + 1) $.

Factor further if possible

Check if the expression can be factored further. The term $ x^2 - 8 $ can be factored as $ (x - 2\sqrt{2})(x + 2\sqrt{2}) $. The term $ x^2 + 1 $ cannot be factored further with real numbers.

Write the final factored form

Combine all the factors to write the final factored form as $ (x - 2\sqrt{2})(x + 2\sqrt{2})(x^2 + 1) $.

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Key Concepts

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

quadratic form

Quadratic form is a way of looking at polynomial expressions that resemble a standard quadratic equation. The standard quadratic equation is of the form $ ax^2 + bx + c $. However, more complex expressions can sometimes be written in a similar form. In our example, the expression $x^4-7x^2-8$ can be seen as a quadratic in terms of $x^2$.

substitution method

The substitution method helps simplify complex expressions. Here, we treat $x^2$ as a single variable. If we let $y = x^2$, the expression $x^4 - 7x^2 - 8$ turns into $y^2 - 7y - 8$. This is a standard quadratic expression, making it easier to factor.

factoring quadratics

Factoring quadratic expressions involves breaking them down into the product of two binomials. For the transformed expression $y^2 - 7y - 8$, we look for numbers that multiply to $-8$ and add to $-7$. These numbers are $-8$ and \1\. Thus, we can write $y^2 - 7y - 8$ as $(y - 8)(y + 1)$.

polynomial factorization

Polynomial factorization extends the simple factorization of quadratics to more complex expressions. After substituting back $x^2$ for \y\, our expression becomes $(x^2 - 8)(x^2 + 1)$. To factor further, we can break down $x^2 - 8$ into $(x-2\root2)(x+2\root2)$. However, $x^2 + 1$ cannot be factored further using real numbers. By combining these steps, the fully factored form is $(x-2\root 2)(x+2\root 2)(x^2+1)$.

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91影视

In the following exercises, factor each expression using any method. $$ x^{4}-7 x^{2}-8 $$

Short Answer

Step by step solution

Identify the quadratic form

Factor the quadratic expression

Substitute back \( x^2 \) for \( y \)

Factor further if possible

Write the final factored form

Key Concepts

quadratic form

substitution method

factoring quadratics

polynomial factorization

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