Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 1-8 x^{2}-9 x^{4} $$

Short answer

-(9x^2 - 1)(x^2 + 1)

Step-by-step solution

- Identify the polynomial

The given polynomial is: 1 - 8x^{2} - 9x^{4}.

- Rearrange the polynomial

Rearrange the polynomial in standard form (highest power of x first): -9x^{4} - 8x^{2} + 1.

- Assume substitution technique

Let's use substitution to simplify. Let y = x^{2}. Replace x^{2} with y: -9y^{2} - 8y + 1.

- Factor the quadratic expression

Now, factor the quadratic -9y^{2} - 8y + 1. To factor it correctly, find two numbers that multiply to (a*c) where a = -9 and c = 1, and add to b = -8.

- Find the numbers

The numbers are -9 and 1, because (-9)*1 = -9 and -9 + 1 = -8.

- Express quadratic in factored form

Use these numbers to express the quadratic: -9y^{2} - 9y + y + 1 = -9y(y+1) + 1(y+1).

- Factor by grouping

Factor by grouping: ( -9y + 1)(y + 1). Don't forget that y = x^{2}.

- Substitute back

Substitute back y with x^{2}: -9(x^{2}) + 1)(x^{2} + 1.

- Final factorized form

The completely factored form is: - (9x^{2} - 1)(x^{2} + 1).

Key concepts

Factorization

Factorization is the process of breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. This is useful in solving polynomial equations because it simplifies complex expressions.
To factor a polynomial, look for common factors, use algebraic identities, or apply factorization techniques like grouping or substitution.
In the exercise provided, the polynomial is factored step-by-step to reveal its simpler components. By recognizing that different terms can be reshaped and combined, we manage to break the complex polynomial into its factors.

Quadratic Equations

A quadratic equation is a polynomial equation of degree 2, generally written as \(ax^2 + bx + c = 0\). Solving quadratic equations often involves factoring, using the quadratic formula, or completing the square.
In our exercise, we transformed part of the polynomial into a quadratic form using a substitution. This method simplified our complex fourth-degree polynomial to a more manageable quadratic form \( -9y^2 - 8y + 1\). Quadratic equations are the backbone of many factorization problems, providing a structured path to solutions.

Substitution Technique

The substitution technique helps simplify complex polynomials by introducing a new variable. This can turn a difficult polynomial into a simpler form that’s easier to factor.
In the given exercise, we set \( y = x^2 \). This reduced our original polynomial \( 1 - 8x^2 - 9x^4 \) to \( -9y^2 - 8y + 1 \). By doing so, the polynomial appears much simpler and falls into the quadratic category, which can be more straightforwardly factored.
Substitution is particularly useful in higher-degree polynomials, as it helps to reduce the complexity temporarily while maintaining the equality of the equations.

Polynomial Rearrangement

Polynomial rearrangement is the process of reordering the terms of a polynomial to make it easier to work with. This often involves writing the polynomial in standard form, which is ordered by decreasing powers of the variable.
In the provided exercise, the polynomial is initially given as \( 1 - 8x^2 - 9x^4 \). The next step places it into standard form: \( -9x^4 - 8x^2 + 1 \). This reordering is crucial because it aligns the terms by degree, which helps identify potential factoring strategies.
Rearrangement is a simple but powerful tool to gain better insights into the structure and possible factorizations of a polynomial.