Problem 44 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ u^{2}-18 u+81 $$

Step-by-Step Solution

Answer

The factored expression is \((u - 9)^2\).

Step 1: Identify the Form of the Quadratic

The expression given is a quadratic trinomial in the form of \( u^2 - 18u + 81 \). The standard form of a quadratic equation is \( ax^2 + bx + c \), where \( a = 1 \), \( b = -18 \), and \( c = 81 \).

Step 2: Determine If Perfect Square Trinomial

Check if this trinomial is a perfect square. A perfect square trinomial takes the form \((x - d)^2\), which equals \( x^2 - 2dx + d^2 \). Here, it's of interest to check:1. \( u^2 \) is already a perfect square.2. Check if 81 is a perfect square. Indeed, \( 81 = 9^2 \).3. Check if \(-18u = 2du\) holds, where \( d = 9 \). Calculate \(-2 \times 9 = -18\).All conditions are satisfied, so the trinomial is a perfect square.

Step 3: Factor the Trinomial

Since the trinomial is a perfect square, it can be factored into the form \((u - 9)^2\).

Step 4: Verify the Factoring

Expand \((u - 9)^2\) to verify:\[(u - 9)(u - 9) = u^2 - 9u - 9u + 81 = u^2 - 18u + 81\].The expression matches the original quadratic, confirming the factoring is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression. It can be written as

• \( (x - d)^2 \)
• or \( (x + d)^2 \)

This means it results from multiplying one binomial by itself. In a perfect square trinomial, both the first term and the last term must be perfect squares.
To check if a trinomial is a perfect square, consider these points:

• The middle term should equal twice the product of the square roots of the first and the last terms.
• For instance, in the expression \( u^2 - 18u + 81 \), both \( u^2 \) and \( 81 \) are perfect squares.
• The middle term, \( -18u \), should equal \(-2 \times 9 \times u\). Here, \( 9 \) is the square root of \( 81 \), and everything checks out.

Practicing recognizing perfect squares can simplify factoring and help avoid lengthy calculations.

Quadratic Trinomial

Quadratic trinomials are expressions that include three terms, usually seen in the standard form as \( ax^2 + bx + c \). These resemble a parabola's algebraic backbone, and they dominate many areas of algebra. Here, \( a \) is the coefficient for the squared term, distinct from the linear \( b \, \text{term}\) and the constant \( c \).
In this form, understanding the structure greatly aids in comprehending how to factor them. For example, the trinomial \( u^2 - 18u + 81 \) neatly fits this structure:

• The coefficient \( a \) is marked by the number preceding \( u^2 \), in this case, \( 1 \).
• \( b \) introduces the factor linked to the single-variable term, which is \( -18 \).
• Then, \( c \), the constant, is set as \( 81 \).

Knowing how to align a trinomial into this familiar shape can make the factoring process less intimidating.

Factoring Process

The factoring process involves breaking down a complex expression into simpler components. These components, when multiplied back together, reconstruct the original expression. Factoring is crucial because it simplifies expressions, making them easier to solve or understand.
The following outlines a typical approach to factoring, particularly for a perfect square trinomial:

• Determine if the quadratic trinomial is indeed a perfect square, which simplifies the factoring immensely.
• If confirmed, proceed by rewriting it in the form \((x - d)^2\) or \((x + d)^2\).
• In our exercise, \( u^2 - 18u + 81 \) verifies as \((u - 9)^2\) since \( 9^2 = 81 \) and the double product matches the middle term.
• It's important to verify this factoring by expanding the expression back out: \((u - 9)^2 = u^2 - 18u + 81\).
• This confirms that our factorization is accurately equivalent to the original expression.

Practicing this method strengthens your ability to tackle even more challenging algebraic equations in the future.