Problem 71 - Algebra A Combined Function

Question

Factor. $$ x^{4}+x^{2}-20 $$

Step-by-Step Solution

Answer

The expression factors to \((x^2 + 5)(x - 2)(x + 2)\).

Step 1: Identify Quadratic Form

The expression provided is a polynomial of degree 4: \( x^4 + x^2 - 20 \). Notice it can be viewed as a quadratic in terms of \( x^2 \) because the exponents of the terms are powers of two. So, consider substituting \( y = x^2 \). This transforms the expression into \( y^2 + y - 20 \).

Step 2: Factor the Quadratic

Next, factor the quadratic expression \( y^2 + y - 20 \). We need two numbers whose product is \(-20\) and sum is \(1\). These numbers are \(5\) and \(-4\). So, the expression factors as \( (y + 5)(y - 4) \).

Step 3: Substitute Back to Original Variable

Now that we have factored in terms of \( y \), we substitute back \( y = x^2 \). Therefore, the expression becomes \( (x^2 + 5)(x^2 - 4) \).

Step 4: Further Factor if Possible

The factor \( x^2 - 4 \) is a difference of squares, which can be further factored. Use the identity \( a^2 - b^2 = (a+b)(a-b) \). Thus, \( x^2 - 4 = (x - 2)(x + 2) \).

Step 5: Write the Complete Factored Form

Now write down the complete factorization using all factored forms: \( x^4 + x^2 - 20 = (x^2 + 5)(x - 2)(x + 2) \).

Key Concepts

Quadratic Form

A quadratic form is a type of polynomial that typically appears in the format \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. It represents any polynomial expression where the greatest exponent of the variable is 2. In the exercise, although starting with a degree 4 polynomial, we identified it could be viewed in quadratic form by treating \( x^2 \) as a single variable.

This simplification allows us to rewrite our expression, \( x^4 + x^2 - 20 \), into the form \( y^2 + y - 20 \) by substituting \( y = x^2 \). This makes it easier to factor, as it transforms a potentially complex degree 4 polynomial into a more manageable quadratic.

• Identify how the expression or part of it can fit this form.
• Perform substitutions to simplify complex polynomials into quadratic form.
• Focus on recognizing patterns of exponents like squares.

Difference of Squares

The difference of squares is a specific algebraic formula, namely \( a^2 - b^2 = (a+b)(a-b) \), used to factor expressions involving two perfect squares with subtraction. This concept becomes handy when you spot terms like \( x^2 - 4 \) in this exercise.

In a difference of squares, you are looking for:

• Two terms that are each perfect squares.
• A subtraction between them.

For example, \( x^2 - 4 \) can be rewritten as \( x^2 - 2^2 \), which fits the form \( a^2 - b^2 \). Using the difference of squares identity, it factors into \( (x - 2)(x + 2) \). This further factorization helps simplify the polynomial even more, and it's an efficient approach to handle specific kinds of polynomials.

Substitution Method

The substitution method is a valuable tool for simplifying polynomial factorization. It involves replacing a complex part of a polynomial with a simpler variable to make the equation easier to handle. In our exercise, this is accomplished by substituting \( y = x^2 \).

This makes handling the polynomial \( x^4 + x^2 - 20 \) feasible by converting it to the quadratic \( y^2 + y - 20 \). The process of substitution simplifies the equation structurally:

• Choose an appropriate substitution to reduce complexity.
• Solve or factor the equation in the new variable.
• Replace back to the original variable after simplification or factorization.

By using proper substitution, intricate higher-degree polynomials can often be broken down into more familiar, easier-to-solve quadratics. It helps in managing equations that may not initially appear straightforward to factor.