Problem 71 - Elementary and Intermediate Algebra

Question

Factor. $$ t^{2}-20 t+100 $$

Step-by-Step Solution

Answer

The expression factors to \((t-10)^2\).

Step 1: Understand the quadratic expression

The given expression is a quadratic trinomial: \( t^{2} - 20t + 100 \). Our task is to factor it into a product of binomials.

Step 2: Identify the structure

Notice that \( t^{2} - 20t + 100 \) resembles the form \( a^2 - 2ab + b^2 \), which can be factored using the perfect square formula: \( (a-b)^2 \).

Step 3: Find the values of 'a' and 'b'

In \( t^{2} \, \), 'a' is 't'. In \( 100 \), which is \( 10^2 \), 'b' is 10. Check that \(-2ab = -20t\) matches by calculating \(-2(t)(10) = -20t\). This confirms 'b' as 10.

Step 4: Apply the perfect square formula

Since both conditions of the perfect square trinomial \( a^2 - 2ab + b^2 \) are met, rewrite the expression as \((t-10)^2\).

Step 5: Verify the factorization

Expand \((t-10)^2\) to ensure it returns the original expression: \((t - 10)(t - 10) = t^2 - 20t + 100\). The factorization is verified.

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a specific type of quadratic trinomial that can be expressed as the square of a binomial. For instance, in the expression \( t^2 - 20t + 100 \), it can be compared to the standard form \( a^2 - 2ab + b^2 \). This form is key in identifying a perfect square trinomial.
To identify a perfect square trinomial:

• Check if the first and last terms are perfect squares.
• Verify the middle term matches \(-2ab\), where \(a\) and \(b\) are the square roots of the first and last terms, respectively.

In our example:

• The first term is \(t^2\) or \(a^2\), so \(a = t\).
• The last term is 100, or \(b^2\), so \(b = 10\).
• The middle term is \(-20t\), which corresponds to \(-2(t)(10)\), verifying that it's a perfect square trinomial.

Thus, the given quadratic can be factored using the formula \((a-b)^2\). This results in \((t-10)^2\).

Quadratic Expressions

A quadratic expression is any polynomial expression of degree two. It typically takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The expression \(t^2 - 20t + 100\) is a classic example of a quadratic expression, where:

• \(a = 1\)
• \(b = -20\)
• \(c = 100\)

Quadratic expressions can often be factored into simpler expressions, particularly when they are perfect square trinomials. Factoring involves rewriting the expression as a product of binomials.
Recognizing patterns such as perfect square trinomials simplifies this process. The ability to convert the quadratic into a simpler form aids in solving related equations and understanding their properties, like determining the vertex or roots of the corresponding quadratic function.

Binomial Factorization

Binomial factorization involves expressing a polynomial as a product of two or more binomial factors. In our case, the polynomial \(t^2 - 20t + 100\) was identified as a perfect square trinomial, which simplified its factorization process.
Once determined to be a perfect square trinomial, we applied the formula \((a-b)^2\), leading to the factorization \((t-10)^2\). This shows that the original expression is the product of two identical binomials, \((t-10)(t-10)\).
This factorization technique can be used to break down other quadratics by recognizing patterns and applying formulas directly, reducing them into their simplest form. Understanding binomial factorization is essential in algebra, allowing you to solve equations and simplify expressions effectively.