Problem 13 - Algebra 1: Concepts and Skills

Question

Choose the correct factorization. $$ x^{2}-10 x+16 $$ $$ \begin{aligned} &a.\quad(x-4)(x-4)\\\ &b.\quad(x-8)(x-2) \end{aligned} $$

Step-by-Step Solution

Answer

The correct factorization option is b. That is, \( (x-8)(x-2) \)

Step 1: Check both options.

The first job is to evaluate both options to see which one, if either, satisfies the condition outlined. Remember that to check a factorization, multiply out the brackets and simplify. The resulting expression should be the original quadratic equation.

Step 2: Check option a.

Let's try out option a. That is \( (x-4)(x-4) \). When this is expanded, it would give \( x^2 - 4x - 4x + 16 = x^2 - 8x + 16 \). This is not the original equation. Therefore, it is not the correct option.

Step 3: Check option b.

Next, let's evaluate option b. It reads \( (x-8)(x-2) \). When multiplied out, this would give \( x^2 - 8x - 2x + 16 = x^2 - 10x + 16 \). This exactly matches the original equation. Therefore, this is the correct option.

Key Concepts

Polynomial Expression

A polynomial expression is a mathematical phrase involving variables, coefficients, and arithmetic operations like addition, subtraction, multiplication, and non-negative integer exponents. It can range from simple to complex, with different terms within the expression. Each term in a polynomial has a degree, which is determined by the exponent on the variable.
For example, the expression given in the problem, \( x^2 - 10x + 16 \), is a polynomial because it includes a square term \( x^2 \), a linear term \( -10x \), and a constant term \( 16 \). Polynomials like this are often encountered in algebra and are foundational for understanding algebraic relationships and problem-solving.

• The terms in a polynomial are separated by addition or subtraction.
• The degree of the polynomial is the highest degree of its terms, which in this case is 2.

Factoring Quadratics

Factoring quadratics is a method used to express a quadratic polynomial in the form \((x - r)(x - s)\), where \(r\) and \(s\) are numbers that make the factorization true. The goal is to find numbers that will both sum up to the coefficient of the linear term and multiply to the constant term.
When factoring \(x^2 - 10x + 16\), we need to determine which pair of numbers adds to \(-10\) (the coefficient of the \(x\) term) and multiplies to \(16\). Through trial and error, or proper calculation, you'd find that \((-8)\) and \((-2)\) are the right numbers. Therefore, the correct factorization is \((x - 8)(x - 2)\), which when expanded, returns the original quadratic.

• Factoring involves finding numbers that satisfy both addition and multiplication conditions.
• This process helps simplify solving roots or solutions of quadratic equations.

Algebraic Multiplication

Algebraic multiplication, especially when dealing with quadratics, involves multiplying expressions to find a single result, often leading to expanding brackets or simplifying an equation. When multiplying out brackets, you apply the distributive property, multiply each term in the first bracket with each term in the second bracket, and add them together.
For example, to check if \((x - 8)(x - 2)\) is the correct factorization for \(x^2 - 10x + 16\), we use algebraic multiplication as follows:

• Multiply \(x\) by \(x\) to get \(x^2\).
• Multiply \(x\) by \(-2\) to get \(-2x\).
• Multiply \(-8\) by \(x\) to get \(-8x\).
• Multiply \(-8\) by \(-2\) to get \(+16\).

This expansion leads to \(x^2 - 8x - 2x + 16\), which simplifies to \(x^2 - 10x + 16\), confirming our factorization is correct.
Algebraic multiplication is essential for verifying factorization and is a fundamental concept for various algebraic operations.