Problem 28 - Precalculus: Functions and Graphs

Question

Factor the polynomial. $$x^{4}-8 x^{3}+16 x^{2}$$

Step-by-Step Solution

Answer

The polynomial is factored as \(x^{2}(x-4)^{2}\).

Step 1: Identify Common Factors

First, we look for any common factors in the polynomial \(x^{4}-8x^{3}+16x^{2}\). Noticing that each term contains an \(x^{2}\), we can factor \(x^{2}\) out of the polynomial. This gives us:\[ x^{2}(x^{2} - 8x + 16) \]

Step 2: Factor the Quadratic Expression

Next, we need to factor the quadratic expression \(x^{2} - 8x + 16\). We look for two numbers that multiply to \(16\) and add up to \(-8\). These numbers are \(-4\) and \(-4\). Hence, the quadratic can be factored as:\[ (x-4)(x-4) \] or \[(x-4)^{2} \]

Step 3: Write the Complete Factorization

Combine the factored quadratic expression with the factored out \(x^{2}\) to write the complete factorization of the original polynomial:\[ x^{2}(x-4)^{2} \]

Key Concepts

Common Factors

Factoring polynomials often starts with identifying what is known as a "common factor." This makes the rest of the factoring process simpler.
An expression can have common factors if all its terms share a particular number or variable.
In our polynomial, \(x^4 - 8x^3 + 16x^2\), each term includes an \(x^2\). Therefore, \(x^2\) is a common factor.
By factoring this out, we simplify the polynomial to \(x^2(x^2 - 8x + 16)\). This step reduces the degree of each term in the polynomial, which is the highest power of \(x\).

• Look for variables or numbers that appear in every term.
• Factor them out to make the equation simpler.

By mastering how to spot and factor out common factors, more complex tasks like complete factorization become manageable.

Quadratic Expression

Once the common factors are taken care of, the next challenge is often a quadratic expression to factor. A quadratic expression is usually in the form \(ax^2 + bx + c\).
In our example, the expression \(x^2 - 8x + 16\) fits the standard quadratic form. The task is to break it down into simpler terms, usually two binomials.
To factor a quadratic expression like this, find two numbers that:

• Multiply to the constant term (here, that's 16).
• Add up to the coefficient of the middle term (here, -8).

For \(x^2 - 8x + 16\), we see that \(-4\) and \(-4\) satisfy both conditions, meaning it factors to \((x-4)(x-4)\) or \((x-4)^2\).
This approach, focusing on multiplication and addition to solve quadratics, is called "factoring by inspection"

Complete Factorization

After identifying common factors and factoring the quadratic expression, the final step of polynomial factoring is writing down the complete factorization.
This involves combining all the pieces together to express the original polynomial as a product of its factors.
In our case, starting with \(x^4 - 8x^3 + 16x^2\), we factored out \(x^2\) from each term, leaving us with \(x^2(x^2 - 8x + 16)\).
Further simplifying the quadratic expression to \((x-4)^2\), the total factorization becomes:
\[x^2(x-4)^2\]
Complete factorization is useful because it gives you a complete breakdown of a polynomial, which can be important for solving equations or understanding the polynomial's properties.
Once you are comfortable with this process, tackling complex mathematical concepts will become easier and more intuitive.