Problem 138 - Introductory Algebra for College Students

Question

Factor completely. $$2 x^{2}-20 x+50$$

Step-by-Step Solution

Answer

The complete factorization of \(2x^{2}-20x+50\) is \((x-5)(2x-10)\)

Step 1: Identifying the Coefficients and Constant

Identify the coefficients and constant in the expression. Here, the coefficient of \(x^{2}\) is 2, the coefficient of \(x\) is -20, and the constant is 50.

Step 2: Find the Product and Factors

Find the product of the coefficient of \(x^{2}\) and the constant term. The product is 2*50 = 100. Now, look for the pair of factors of 100 that add up to the coefficient of x, which is -20. The pair here is -10 and -10 because -10*-10 = 100 and -10+(-10) = -20.

Step 3: Rewrite the Polynomial

Now, rewrite the given polynomial replacing the middle term with the factors of the product. The expression \(2x^{2}-20x+50\) becomes \(2x^{2}-10x-10x+50\).

Step 4: Factor by Grouping

Factor by grouping. Group the first two terms and the last two terms separately. By factoring out \(2x\) from the first group and -10 from the second group, the equation becomes \(2x(x-5) - 10(x-5)\).

Step 5: Factor out the Common Factor

Now, look for a common factor in both groups. Here, \(x-5\) is the common factor. Factor it out to yield the final factored polynomial: \((x-5)(2x-10)\).

Key Concepts

Polynomial expressions

Polynomial expressions are mathematical expressions involving variables raised to whole number powers and coefficients. Understanding them is a foundational step in algebra. A polynomial may have one or more terms, and each term consists of a coefficient and one or more variables raised to a power. For example, in the expression \(2x^2 - 20x + 50\), there are three terms:

• \(2x^2\) - the quadratic term where 2 is the coefficient.
• \(-20x\) - the linear term where -20 is the coefficient.
• \(50\) - the constant term.

Each polynomial term's degree is determined by the power of the variable. The highest power in a polynomial gives its degree. For instance, \(2x^2\) makes this a quadratic polynomial because the highest power is 2.
Understanding the structure of polynomial expressions allows for the application of different algebraic techniques, such as factoring.

Factor by grouping

Factor by grouping is a powerful algebraic technique used to factorize polynomials. It involves arranging a polynomial expression into groups that can be factored separately.
Here's how to factor by grouping using the example \(2x^2 - 20x + 50\):

• First, split the middle term (-20x) into two terms that can help in grouping. In this example, \(-10x\) and \(-10x\) are used because their product is equivalent to 100 and their sum is -20.
• Rewrite the polynomial as \(2x^2 - 10x - 10x + 50\).
• Group the terms: \((2x^2 - 10x) + (-10x + 50)\).
• Factor each group individually: from the first group \(2x(x - 5)\), and from the second group \(-10(x - 5)\).
• Notice that \((x - 5)\) is a common factor for both groups. Factor it out: \((x - 5)(2x - 10)\).

Factoring by grouping requires careful selection of terms and can simplify complex polynomial expressions efficiently.

Algebraic techniques

Algebraic techniques are strategies used to simplify and solve polynomial expressions. They are essential in solving equations and analyzing mathematical relationships.
Some key algebraic techniques include:

• Factoring: Breaking down expressions into simpler, multiplicative components, like turning \(2x^2 - 20x + 50\) into \((x - 5)(2x - 10)\).
• Distributive Property: Using a\(b + c = ab + ac\) helps in expanding and factoring expressions.
• Combining Like Terms: Simplifying expressions by merging terms with the same variable parts, essential for maintaining clarity in expressions.

Each technique has a role in managing expressions and solving equations, providing tools for problem-solving in both simple and complex scenarios. Mastery of these methods develops a stronger understanding of algebra and its applications.