In Exercises 103–114, factor completely. 6x⁴+35x²−6

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1. Identify the structure of the polynomial. Notice that the given polynomial, 6x⁴ + 35x² − 6, is a quadratic in form because the powers of x are multiples of 2 (x⁴ and x²). Let y = x² to rewrite the polynomial as 6y² + 35y − 6.
2. Factor the quadratic expression 6y² + 35y − 6. Look for two numbers that multiply to the product of the leading coefficient (6) and the constant term (−6), which is −36, and add to the middle coefficient (35). These numbers are 36 and −1.
3. Rewrite the middle term (35y) using the two numbers found: 6y² + 36y − y − 6. Group the terms into two pairs: (6y² + 36y) and (−y − 6).
4. Factor out the greatest common factor (GCF) from each group. From the first group, factor out 6y, and from the second group, factor out −1: 6y(y + 6) − 1(y + 6).
5. Factor out the common binomial factor (y + 6): (6y − 1)(y + 6). Replace y with x² to return to the original variable: (6x² − 1)(x² + 6). This is the completely factored form of the polynomial.

Key Concepts

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler polynomial factors. This process is essential for simplifying expressions and solving equations. Common techniques include finding the greatest common factor (GCF), using special products like the difference of squares, and applying methods such as grouping or the quadratic formula when applicable.

Quadratic Form

The expression 6x^4 + 35x^2 - 6 can be rewritten in a quadratic form by substituting x^2 with a new variable, say y. This transforms the polynomial into a standard quadratic equation, making it easier to factor. Recognizing and utilizing this substitution is crucial for simplifying higher-degree polynomials.

Zero Product Property

The Zero Product Property states that if the product of two or more factors equals zero, at least one of the factors must be zero. This principle is vital when solving polynomial equations after factoring, as it allows us to find the roots or solutions of the equation by setting each factor equal to zero.