Problem 70 - Introductory Algebra for College Students

Question

Factor completely: \(2 x^{3}-20 x^{2}+50 x .\) (Section 6.5 Example 2 )

Step-by-Step Solution

Answer

The equation \(2x^3 - 20x^2 + 50x\) factorizes to \(2x(x-5)^2\).

Step 1: Identify Common Factors

Before factoring, it's important to check for any common factors. By examining the coefficients in the equation \(2x^3 - 20x^2 + 50x\), we can see the common factor is '2x'. Factoring this out, we have \(2x (x^2 - 10x + 25) \).

Step 2: Factorize Quadratic Inside Brackets

To further factorize, we need to break down the quadratic equation. The equation within the brackets can be written in the form of \(ax^2 - bx + c\), where a = 1, b = 10 and c = 25. The quadratic expression is a perfect square if it can be written in the form of \((px + q)^2\), where 'p' is the square root of 'a' and 'q' is the square root of 'c'. The expression inside the brackets fits this description perfectly, because the coefficients are perfect squares and can be written as \((x-5)^2\).

Step 3: Combine Terms for Final Answer

Combining the results from the previous steps, the completely factored form of the equation is \(2x(x-5)^2\).

Key Concepts

Common Factors in Algebra

Understanding how to spot and extract common factors is essential in simplifying algebraic expressions and equations. When factoring polynomials, the first thing you look for is a common factor, which is a term that all parts of the polynomial share. Take the example from the exercise: the polynomial is given as 2x^3 - 20x^2 + 50x. Here, each term is divisible by 2x, making it the common factor. We can then factor 2x out of the polynomial, simplifying the expression to 2x(x^2 - 10x + 25). This process is important because it often makes the remainder of the problem simpler and can reveal further factorization opportunities, as we see with the resultant quadratic expression.
In general, identifying common factors allows you to write a polynomial as the product of simpler polynomials, leading you a step closer to solving or simplifying the algebraic expression at hand. Searching for the greatest common factor (GCF) can help break down more complex problems into manageable pieces, aiding in the solution of equations and simplification of expressions.

Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic polynomial that takes the form (ax)^2 + 2abx + b^2 where the first and last terms are perfect squares, and the middle term is twice the product of the square roots of those terms. In simpler terms, a perfect square trinomial can always be broken down into a binomial multiplied by itself, such as (px + q)^2.

Looking again at our exercise, once the common factor 2x is removed, we are left with the trinomial x^2 - 10x + 25. With a little observation, we can see that x^2 is the square of x, and 25 is the square of 5. Furthermore, the middle term, -10x, is twice the product of the square roots of x^2 and 25, which confirms that the trinomial is indeed a perfect square. It can then be factored into (x-5)^2. This insight greatly simplifies solving the problem and is a common technique used to factorize quadratic expressions.

Quadratic Expressions

Quadratic expressions are second-degree polynomials, generally taking the form ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. Quadratics often pop up in various areas of mathematics and are foundational in the study of algebra.

Factoring is a crucial skill when working with quadratic expressions, as it allows one to simplify expressions and solve quadratic equations. The factoring process involves rewriting the quadratic in the form of a product of two binomials or finding perfect square trinomials. Sometimes, as seen in the solution to the exercise, the quadratic expression can be a perfect square, allowing you to apply patterns recognized from the difference of squares or perfect square formulas. In other instances, the AC method or the quadratic formula can be used when the expression does not easily factor into simpler equations. Being adept at recognizing the forms of quadratic expressions and understanding the different methods of factoring them is vital for students to solve algebraic problems efficiently.