Problem 21 - Algebra A Combined Function

Question

Factor each trinomial completely. If a polynomial can't be factored, write "prime." $$ a^{2}-10 a b+16 b^{2} $$

Step-by-Step Solution

Answer

The trinomial factors to \((a - 2b)(a - 8b)\).

Step 1: Identify the Trinomial Form

The given trinomial is expressed as \( ax^2 + bx + c \), where we need to factor \( a^2 - 10ab + 16b^2 \). Here, it can be viewed as a quadratic in terms of the variable \( a \), so \( a = 1 \), \( b = -10b \), and \( c = 16b^2 \).

Step 2: Find Factors of c That Add to b

We need two numbers that multiply to \( 16 \cdot b^2 \) and add to \(-10b\). The factors that work are \(-2b\) and \(-8b\), since \((-2b)\cdot(-8b) = 16b^2\) and \((-2b) + (-8b) = -10b\).

Step 3: Rewrite the Middle Term

Express \(-10ab\) as a sum of \(-2ab\) and \(-8ab\). Thus, the trinomial \( a^2 - 10ab + 16b^2 \) rewrites to \( a^2 - 2ab - 8ab + 16b^2 \).

Step 4: Factor by Grouping

Group the terms to factor by grouping: \((a^2 - 2ab) + (-8ab + 16b^2)\). Factor each group separately: \(a(a - 2b) - 8b(a - 2b)\).

Step 5: Factor Out the Common Binomial

Since \( (a - 2b) \) is common, factor it out: \((a - 2b)(a - 8b)\).

Step 6: Confirm Complete Factorization

Check if the factors \((a - 2b)(a - 8b)\) multiply back to the original trinomial. They do, confirming the factorization is complete.

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial that is of the second degree, meaning the highest power of the variable involved is 2. Trinomials, which consist of three terms, often appear in the form of a quadratic expression, given by \( ax^2 + bx + c \). Here, the coefficients \( a \), \( b \), and \( c \) are constants. In the exercise, the trinomial \( a^2 - 10ab + 16b^2 \) can be viewed as a quadratic equation in terms of \( a \).

In this context:

• \( a \) is the term containing the highest power of variable \( a \).
• \( b \) is the coefficient of the middle term, often involving two variables.
• \( c \) is the constant term or the term without \( a \), here involving the variable \( b^2 \).

Quadratic expressions are central to algebra, and factoring them helps in simplifying and solving equations. The goal of factoring is to express the quadratic as a product of two binomials, which makes solving or simplifying the expression much easier.

Factor by Grouping

Factor by grouping is a method used in polynomial factorization, especially effective for simplifying trinomials. The main idea is to rearrange the terms so that they can be grouped into pairs which allow you to factor out a common element from each pair.

For the trinomial \( a^2 - 10ab + 16b^2 \), once the middle term is rewritten as \(-2ab - 8ab\), the expression can be grouped into two pairs: \((a^2 - 2ab) + (-8ab + 16b^2)\).
This method involves:

• Identifying pairs of terms that have a common factor.
• Factoring out the greatest common factor (GCF) from each pair.
• The expression becomes \(a(a - 2b) - 8b(a - 2b)\).
• Since \( (a - 2b) \) is now a common binomial factor, it can be factored out.

Factor by grouping simplifies the trinomial into the product of two binomials, which is easier to handle and verify. This technique is particularly useful when direct factoring seems complicated or non-intuitive.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as the product of its factors. It involves breaking down complex polynomials into simpler, multiplicative components, which are usually of lower degrees.

The exercise provided illustrates the principle of factorization using a trinomial. The original trinomial \( a^2 - 10ab + 16b^2 \) is reduced to \((a - 2b)(a - 8b)\) through factorization.
Steps involved in polynomial factorization include:

• Identifying the type of polynomial – here it's a trinomial, a quadratic in terms of one variable.
• Using methods such as trial and error, factoring by grouping, or special factor formulas to find the factors.
• Checking if the factored form is correct by expanding the factors to see if they multiply back to the original polynomial.

Effective factorization simplifies solving equations, helps in finding roots, and aids in understanding the underlying structure of the polynomial. For students, practicing factoring makes them better equipped to tackle more complex algebraic problems in their education journey.