Problem 83 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$x^{4}+6 x^{2}+9$$

Step-by-Step Solution

Answer

The polynomial factors completely as \((x^2 + 3)^2\).

Step 1: Identify the Structure

First, observe the given polynomial: \(x^4 + 6x^2 + 9\). Notice that the exponents of \(x\) are multiples of 2. This suggests it might be a perfect square trinomial.

Step 2: Recognize Perfect Square Trinomial Form

A perfect square trinomial has the form \((ax^2 + bx + c)^2 = a^2x^4 + 2abx^2 + b^2\). Observe that the polynomial \(x^4 + 6x^2 + 9\) matches the form \((x^2 + 3)^2 = x^4 + 6x^2 + 9\), which is a perfect square trinomial.

Step 3: Factor the Trinomial

Since \(x^4 + 6x^2 + 9\) is a perfect square trinomial, it can be factored as \((x^2 + 3)^2\).

Step 4: Check and Simplify

Verify the factorization by expanding \((x^2 + 3)^2\). Expand it to get \(x^2 + 3)(x^2 + 3) = x^4 + 6x^2 + 9\), which matches the original polynomial. As there are no linear or reducible quadratic factors, the factorization \((x^2 + 3)^2\) is simplified and complete.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial. It takes the form of \(a^2x^4 + 2abx^2 + b^2\). Understanding this structure helps in recognizing and factoring polynomials quickly.
An important aspect is that when you square a binomial of the form \(a + b\), you end up with a perfect square trinomial:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• Similarly, \(x^4 + 6x^2 + 9\) matches the form \( (x^2 + 3)^2\).

Once you can identify this pattern, your task simplifies to expressing the trinomial as the square of a binomial. This is what makes understanding perfect square trinomials valuable in factorization.

Exponents

Exponents tell you how many times a number, known as the base, is multiplied by itself.
They play a crucial role in polynomial expressions and their factorizations. A key feature of the polynomial \(x^4 + 6x^2 + 9\) is the presence of exponents that are all multiples of two.
In this case, the exponents indicate a likelihood of the polynomial being rewritten in simpler terms.

• The exponent \(x^4\) suggests two layers of squaring: \(x^4 = (x^2)^2\).
• The term \(x^2\) is also a square of \(x\).

Recognizing this symmetry in exponents can help identify underlying patterns such as perfect squares, facilitating easier factorization.

Factorization Techniques

Factorization is breaking down a complex expression into simpler factors. Several techniques help simplify this process.
For example, recognizing perfect square trinomials, as we did in the expression \(x^4 + 6x^2 + 9\), is a useful technique:

• First, observe the form of the polynomial to determine if it can be rewritten as a binomial squared.
• Apply patterns like \(a^2 + 2ab + b^2\) to break down the expression easily.

Additionally, make sure to expand the factors post factorization to ensure they return to the original polynomial.
An accurate factorization is key in expressing and simplifying polynomials.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically written as \(ax^2 + bx + c\). They appear in various forms, and sometimes like in our example, even as hidden squares.
The polynomial \(x^4 + 6x^2 + 9\) does not initially look like a simple quadratic.
However, when we observe the pattern \( (x^2 + 3)^2 \), we effectively reduce it to a simpler quadratic form, \(x^2 + 3\), squared.
Recognizing that more complex polynomials may disguise simpler quadratic forms is crucial.

• By treating higher-degree expressions, such as \(x^4 + 6x^2 + 9\), we look for underlying quadratic patterns.
• This simplifies not only factorization but any calculations or integrations involving the expression.

Through mastering quadratic expressions, you gain better control over polynomial functions.