Problem 55 - College Algebra

Question

For the following exercises, factor the polynomials completely. $$ 16 x^{4}-200 x^{2}+625 $$

Step-by-Step Solution

Answer

The polynomial factors completely as \((2x - 5)^2(2x + 5)^2\).

Step 1: Identify the polynomial form

Observe the given polynomial: \(16x^4 - 200x^2 + 625\). Notice that it can be expressed as a quadratic in form by considering \(u = x^2\), resulting in \(16u^2 - 200u + 625\). This is now a quadratic polynomial in terms of \(u\).

Step 2: Check for perfect square trinomial

Check if the quadratic \(16u^2 - 200u + 625\) is a perfect square trinomial. A quadratic \(au^2 + bu + c\) is a perfect square if it matches the form \((mu - n)^2 = m^2u^2 - 2mnu + n^2\). Compare the given equation and notice \(c = 625 = n^2\), suggesting \(n = 25\).

Step 3: Verify factorization

Continue verifying the perfect square form by checking if \(-200u = -2 \times 4 \times 25 \times u\). We find that the polynomial can be rewritten as \((4u - 25)^2.\)

Step 4: Substitute back the original variable

Reverse the earlier substitution where \(u = x^2\). Thus, \((4u - 25)^2 = (4x^2 - 25)^2\).

Step 5: Factor further using difference of squares

Recognize \(4x^2 - 25\) as a difference of squares: \((2x)^2 - (5)^2 = (2x - 5)(2x + 5)\). Thus, \((4x^2 - 25)^2\) becomes \([ (2x - 5)(2x + 5)]^2 = (2x - 5)^2(2x + 5)^2\).

Step 6: Write the final factored form

The complete factorization of the polynomial is \((2x - 5)^2(2x + 5)^2\).

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a specific type of quadratic expression. It looks like what you'd get after squaring a binomial. So, if you have a binomial expression such as

• (a + b)

and you square it, the result is a perfect square trinomial:

• (a + b)^2 = a^2 + 2ab + b^2

For example, consider the quadratic expression \(16u^2 - 200u + 625\). This can be a perfect square trinomial if it fits the pattern

• (m\times u - n)^2 = m^2u^2 - 2mnu + n^2

In this exercise, by comparing coefficients with

• \(c = 625 = n^2\)

this hints that perhaps our quadratic can be expressed as a perfect square. Hence, finding that \(n = 25\), the polynomial is rewritten as

• \((4u - 25)^2\)

Knowing how to match the terms of a quadratic to this format is crucial for completing perfect square trinomial problems.

Quadratic Polynomial

A quadratic polynomial is a type of polynomial characterized by its degree, which is 2. It generally takes the form

• ax^2 + bx + c

where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable. In this exercise, the given polynomial is

• \(16x^4 - 200x^2 + 625\).

We can observe that it takes the form of a quadratic polynomial in disguise when considering

• \(u = x^2\)

which simplifies our polynomial to

• \(16u^2 - 200u + 625\).

By transforming the variable, we can easily work with it as a typical quadratic polynomial while still solving the original context. Understanding this fundamental concept allows you to recognize similar forms in complex problems and make them simpler to solve.

Difference of Squares

The difference of squares is another important pattern in algebra that makes factoring certain expressions much easier. An expression that looks like this

• a^2 - b^2

is a difference of squares and can be factored as

• (a - b)(a + b)

This formula is quite useful, especially in our exercise with

• \(4x^2 - 25\),

because it can be recognized as such. The polynomial simplifies to

• (2x)^2 - (5)^2

which fits the difference of squares pattern perfectly. Once identified, it becomes clearer how to factor

• (4x^2 - 25)\

into

• (2x - 5)(2x + 5).

This step is essential in breaking down complex expressions by recognizing and applying simple formulas.