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Chapter 6: Problem 129

In the following exercises, foctor using substitution. $$ x^{4}-3 x^{2}-28 $$

Short Answer

Expert verified
\( (x^2 - 7)(x^2 + 4) \)

Step by step solution

01

Identify substitution variable

To make factorization easier, identify a substitution variable. Let \( u = x^2 \). This substitution simplifies the given expression.
02

Substitute and rewrite the expression

Using the substitution \( u = x^2 \), rewrite \( x^4 - 3x^2 - 28 \) as \( u^2 - 3u - 28 \).
03

Factor the quadratic expression

Now, factor \( u^2 - 3u - 28 \). Look for two numbers that multiply to -28 and add to -3. These numbers are -7 and 4. Hence, \( u^2 - 3u - 28 \) can be factored as \( (u - 7)(u + 4) \).
04

Substitute back the original variable

Replace \( u \) with \( x^2 \) in the factored expression. Therefore, \( (u - 7)(u + 4) \) becomes \( (x^2 - 7)(x^2 + 4) \).

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Key Concepts

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

Understanding Quadratic Expressions in Algebra
A quadratic expression is a polynomial of degree 2. In simpler terms, it's an expression where the highest power of the variable (usually denoted as x) is 2. An example of a standard quadratic expression is: \[ ax^2 + bx + c \] where \( a \), \( b \), and \( c \) are constants. Quadratic expressions often appear in various algebraic problems and knowing how to manipulate them is crucial. Quadratics are usually solved through methods like factoring, using the quadratic formula, or completing the square. These methods offer different ways to find the roots or solutions of the quadratic expressions.
The Substitution Method in Algebra
The substitution method is a powerful tool to simplify complex algebraic expressions. By replacing a part of the expression with a different variable, you can make it easier to manage. In our exercise, we substituted\( u \) for\( x^2 \). This step transforms a more complicated polynomial into a simpler quadratic one. Here鈥檚 the process:
  • Identify which part of the expression to substitute.
  • Replace this part with a new variable (e.g.,\( u \)).
  • Simplify and solve the new expression.
  • Substitute the original variable back in.
This method is particularly useful when dealing with higher degree polynomials, as it reduces the effort needed to factorize or solve them.
Polynomial Factorization Explained
Polynomial factorization involves breaking down a polynomial into simpler terms (factors) that, when multiplied, give the original polynomial. For quadratic expressions, like the one we鈥檙e working with (\( x^4 - 3x^2 - 28 \)), the procedure includes these steps:
  • Rewrite the expression using substitution (if necessary).
  • Find two numbers that multiply to the constant term and add up to the coefficient of the linear term.
  • Use these numbers to split the middle term and then factor by grouping.
In our example, substituting \( x^2 \) with \( u \) makes it easier: we work with \( u^2 - 3u - 28 \), which factors nicely into \( (u - 7)(u + 4) \). Remember to substitute back afterwards to return to the original variables. Practicing factorization with various polynomials will help you become more confident and efficient at breaking them down, an essential skill for algebra.

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