Problem 92 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 6 w^{8}+17 w^{4}+12 $$

Step-by-Step Solution

Answer

The polynomial factors as \((2w^4 + 3)(3w^4 + 4)\).

Step 1: Recognize a Quadratic Form

Notice that the polynomial \(6w^{8} + 17w^{4} + 12\) can be seen as a quadratic in terms of \(w^{4}\) by substituting \(u = w^{4}\). This gives us \(6u^{2} + 17u + 12\).

Step 2: Write in Standard Quadratic Form

The polynomial is already in the standard form \(6u^2 + 17u + 12\). Here, \(a = 6\), \(b = 17\), and \(c = 12\).

Step 3: Factor the Quadratic Expression

We need to find two numbers that multiply to \(a \cdot c = 6 \cdot 12 = 72\) and add to \(b = 17\). These numbers are 8 and 9. Rewriting \(17u\) as \(8u + 9u\), we have \(6u^2 + 8u + 9u + 12\).

Step 4: Group and Factor by Grouping

Group the terms: \((6u^2 + 8u) + (9u + 12)\). Factor each group: \(2u(3u + 4) + 3(3u + 4)\).

Step 5: Factor Common Binomial

Since \(3u + 4\) is common, factor it out: \((2u + 3)(3u + 4)\).

Step 6: Substitute Back in Terms of w

Replace \(u\) with \(w^4\) to get \((2w^4 + 3)(3w^4 + 4)\) as the factorization of the original polynomial.

Key Concepts

Quadratic Form

Understanding the quadratic form is crucial when working with polynomial expressions, especially higher degree ones like the example given. A quadratic form is typically expressed in the form of a second-degree polynomial, which takes a shape similar to \(ax^2 + bx + c\). However, when dealing with more complex polynomials, we sometimes "transform" them into this standard quadratic form by making a clever substitution.
In the provided example, we have the polynomial \(6w^8 + 17w^4 + 12\). By recognizing the pattern, we can simplify it into a quadratic form using a substitution: let \(u = w^4\). This transforms the expression into \(6u^2 + 17u + 12\). Now, it fits nicely into the quadratic template.

• The purpose of this substitution is to simplify the factoring process.
• Once factored, we substitute back to find the factors in terms of the original variable.

Be on the lookout for such opportunities to simplify complex polynomial expressions by identifying hidden quadratic forms.

Factorization

Factorization is the process of breaking down complex expressions into simpler components, or 'factors,' that multiply together to obtain the original expression. When it comes to polynomials, factorization is a powerful tool that often involves finding binomial pairs or using techniques like grouping.
In the step-by-step solution of our previous polynomial example, after substituting \(u = w^4\), we end up with a familiar quadratic form \(6u^2 + 17u + 12\). To factor this expression, we need to identify two numbers that multiply to the product of the leading coefficient \(a = 6\) and the constant term \(c = 12\), resulting in \(72\), and at the same time add up to the middle coefficient \(b = 17\). These numbers are found to be 8 and 9.

• This allows us to rewrite the middle term and then employ a strategy called grouping, which means partitioning the expression into parts that share a common factor.
• By grouping terms and factoring each group, we identify a shared binomial factor, leading us to the complete factorization.

Factorization is a logic puzzle that, once solved, tells us more about the roots or solutions of the polynomial equation.

Polynomial Expressions

Polynomial expressions are mathematical phrases that can include variables, numbers, and operations like addition, subtraction, multiplication, and exponentiation by a non-negative integer. They are foundational in algebra and useful across many areas of mathematics.
Polynomials are characterized by their degree, which is determined by the highest power of the variable present. In our example, while it might initially seem highly complex due to the degree of 8, recognizing patterns can simplify the expression into forms that are easier to work with, such as a quadratic.

• The original expression \(6w^8+17w^4+12\) is a polynomial of degree 8, indicative of its highest exponent on the variable \(w\).
• By redefining the expression through substitution (\(u = w^4\)), it transforms into a polynomial of degree 2 in terms of \(u\), simplifying our analysis and solution.

Understanding how to work with polynomial expressions, recognizing when to apply substitutions, and employing factorization are key skills that make handling high-degree polynomials more approachable and less intimidating.