Problem 151 In the following exercises, fact... [FREE SOLUTION]

Chapter 6: Problem 151

In the following exercises, factor each expression using any method. $$ x^{4}-4 x^{2}-12 $$

Short Answer

Expert verified

The factored form is \[ (x^{2} - 6)(x^{2} + 2) \].

Step by step solution

Recognize the polynomial structure

Identify that the given polynomial is a quadratic expression in terms of a variable substitution. Let鈥檚 write the polynomial in a more familiar form by setting a new variable: Let \[ u = x^{2} \] so that the expression becomes \[ u^{2} - 4u - 12 \].

Factor the quadratic expression

The expression \[ u^{2} - 4u - 12 \] needs to be factored. Look for two numbers that multiply to -12 (the constant term) and add up to -4 (the coefficient of the middle term). These numbers are -6 and 2. Hence, the expression factors into: \[ (u - 6)(u + 2) \].

Substitute back the original variable

Replace the variable \[ u \] with \[ x^{2} \]. This gives the following factored form: \[ (x^{2} - 6)(x^{2} + 2) \].

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Key Concepts

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

quadratic expressions

A quadratic expression is a polynomial of degree 2. This means the highest exponent of the variable is 2.
Quadratic expressions are typically written in the form ax^2 + bx + c, where a, b, and c are constants.
An example of a quadratic expression is x^2 - 4x - 12.
Quadratics can be factored, solved using the quadratic formula, or completed the square. Factoring means rewriting the expression as a product of two binomials or simpler expressions.

variable substitution

Variable substitution is a helpful technique to simplify complicated expressions.
It involves replacing a part of an expression with a new variable.
Consider the expression x^4 - 4x^2 - 12. We see that x^4 can be written as (x^2)^2, suggesting substituting u = x^2.
Now our expression becomes u^2 - 4u - 12.
This technique can make complex polynomial problems more manageable by transforming them into simpler forms, like quadratic expressions.

polynomial factorization

Polynomial factorization involves expressing a polynomial as a product of its factors.
Factors are polynomials of lower degrees that multiply together to give the original polynomial.
In our example, we need to factor the quadratic expression u^2 - 4u - 12.
We look for two numbers that both multiply to give the constant term (-12) and add up to the coefficient of the linear term (-4).
These numbers are -6 and 2, which means we can rewrite the quadratic as (u - 6)(u + 2).
Substituting back u = x^2, we get (x^2 - 6)(x^2 + 2).
This is the factored form of the original polynomial, illustrating how polynomial factorization breaks down complex expressions into more manageable pieces.

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91影视

In the following exercises, factor each expression using any method. $$ x^{4}-4 x^{2}-12 $$

Short Answer

Step by step solution

Recognize the polynomial structure

Factor the quadratic expression

Substitute back the original variable

Key Concepts

quadratic expressions

variable substitution

polynomial factorization

One App. One Place for Learning.

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