Problem 60 - Introductory Algebra for College Students

Question

Factor completely. $$x^{4}-22 x^{3}+120 x^{2}$$

Step-by-Step Solution

Answer

The complete factorization of the expression \(x^{4} - 22x^{3} + 120x^{2}\) is \(x^{2}(x-2)(x-20)\).

Step 1: Looking for Common Factors

Verify if there are common factors among all the terms in the polynomial \(x^{4}-22x^{3}+120x^{2}\). If there are, factor them out. However, in this case, there are no common factors other than 1.

Step 2: Rearranging the Polynomial

Rearrange the polynomial by separating the terms in a way that allows the application of binomial factoring techniques. \(x^{4}-22x^{3}+120x^{2} = x^{2}(x^{2}-22x+120)\).

Step 3: Factoring the Quadratic Term

Factoring the quadratic expression \(x^{2}-22x+120\), we are looking for two numbers that multiply to 120 and add up to -22. These numbers are -2 and -20, so we rewrite the expression as \(x^{2}(x-2)(x-20)\)

Key Concepts

Common Factors

When factoring polynomials, the first step is often to look for common factors. This means finding a number or variable that each term in the polynomial shares. By factoring out this common element, you simplify the polynomial to make further operations easier. In the polynomial \(x^4 - 22x^3 + 120x^2\), we determine early on that the greatest common factor (GCF) for these terms is \(x^2\). Why is this important?

• Finding common factors can reduce the complexity of the polynomial.
• It prepares the polynomial for further factoring, revealing simpler hidden patterns.
• It helps in rewriting the polynomial in a more manageable form.

Choosing \(x^2\) as the GCF from all terms, you factor it out, allowing for the remaining polynomial \((x^2 - 22x + 120)\) to be analyzed at a simpler level.

Polynomial Rearrangement

Once common factors are identified and factored out, rearranging the polynomial can make it easier to factor further. For example, you may reassemble terms in a way that reveals binomial or trinomial structures. With our polynomial \(x^4 - 22x^3 + 120x^2\), once we factor out \(x^2\), we are left with the expression \(x^2(x^2 - 22x + 120)\).

• Rearranging often involves rewriting the terms in form of a recognizable polynomial equation, like quadratics.
• This process supports further factoring techniques, like splitting the middle term or using the quadratic formula.
• It enables easier identification of patterns, like perfect square trinomials or difference of squares.

Correct rearrangement is a vital step in seeing what simple factoring methods can be applied next to continue reducing the expression.

Quadratic Expressions

Quadratic expressions typically have the form \(ax^2 + bx + c\), where you can often apply specific techniques to factor them into simpler binomial expressions. The quadratic expression in our exercise is \(x^2 - 22x + 120\). To factor this type of expression, follow these steps:

• Identify two numbers that multiply to \(c\) (the constant term) and sum to \(b\) (the coefficient of \(x\)).
• For \(-22x\) and \(120\), the numbers \(-2\) and \(-20\) satisfy these requirements since \(-2 \times -20 = 120\) while \(-2 + (-20) = -22\).
• Rewrite the quadratic as \((x - 2)(x - 20)\), demonstrating that \((x^2 - 22x + 120)\) can be factored into two binomials.

This systematic approach simplifies complex quadratic expressions step by step, leading to easily understandable solutions.