Problem 54 - Intermediate Algebra

Question

Simplify. Assume that the variables represent any real number. $$\sqrt{x^{2}-8 x+16}$$ (Hint: Factor the polynomial first.)

Step-by-Step Solution

Answer

The simplified form is \(|x-4|\).

Step 1: Identify the Expression

We are given the expression \(\sqrt{x^{2}-8x+16}\). To simplify it, the first step is to look inside the square root and focus on the polynomial \(x^{2} - 8x + 16\).

Step 2: Factor the Polynomial

Notice that the polynomial \(x^{2} - 8x + 16\) can be factored. Look for two numbers that multiply to 16 (the constant term) and add up to -8 (the coefficient of x). These numbers are -4 and -4. Therefore, the polynomial can be written as \((x-4)(x-4)\) or \((x-4)^2\).

Step 3: Simplify Inside the Square Root

With the polynomial factored, the expression inside the square root becomes \((x-4)^2\). The square root of a squared term \((x-4)^2\) simplifies to \(|x-4|\), which means the absolute value of \(x-4\).

Step 4: Final Simplification

The original expression \(\sqrt{x^{2}-8x+16}\) thus simplifies to \(|x-4|\). This means the result depends on whether \(x\) is greater than, equal to, or less than 4, but as an expression, it is \(|x-4|\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a process where you express the polynomial as a product of its simpler components called factors. When simplifying radical expressions, especially those under the square root, factoring can often make the task easier and more straightforward.
For example, consider the quadratic polynomial given in the exercise:

• It is of the form \(x^2 - 8x + 16\).
• To factor this polynomial, identify two numbers whose product is the constant term (16) and whose sum is the linear coefficient (-8).
• In this case, both numbers are -4, so the polynomial factors into \((x - 4)(x - 4)\) or \((x-4)^2\).

Factoring into \((x-4)^2\) is particularly useful because it directly relates to taking square roots, simplifying the next steps in the exercise. Recognizing and utilizing factorable patterns such as these is crucial in algebra.

Square Root

The square root is a mathematical operation that finds the original number which, when multiplied by itself, gives the value under the root. In this example, we focus on how the square root interacts with squares.

• When you have a square root of a squared expression, like \(\sqrt{(x-4)^2}\), it simplifies based on the properties of the square and square root functions.
• In simple terms, \(\sqrt{a^2} = |a|\), where \(|a|\) signifies the absolute value of \(a\).

This is because the square root and square are inverse functions, effectively canceling each other. The absolute value comes into play since square roots only deal with non-negative values, ensuring the final simplified result remains non-negative as well.

Absolute Value

The absolute value function, denoted as \(|x|\), requires one to consider only the non-negative form of a number. It measures the "distance" from zero on a number line, ignoring whether this distance is to the left or right of zero.

• In our problem, the simplification \(\sqrt{(x-4)^2} = |x-4|\) applies absolute value because the square root of a square demands non-negative outcomes, irrespective of whether \(x-4\) is positive or negative.
• This means \(|x-4|\) is always positive or zero, depending on the value of \(x\). For example:
  • If \(x > 4\), the result is \(x-4\).
  • If \(x < 4\), the result flips to \(4-x\).
  • If \(x = 4\), the result is \(0\).

Understanding absolute value helps in preserving the mathematical correctness of simplified expressions and is a key piece of many algebraic solutions.