Problem 63 - Elementary Algebra

Question

For the following problems, factor the polynomials, if possible. $$ 2 x^{2}+6 x-20 $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial \(2x^2+6x-20\) is \(2(x+5)(x-2)\).

Step 1: Identify the greatest common factor (GCF)

First, we need to find the GCF of the terms in the polynomial. The terms are \(2x^2\), \(6x\), and \(-20\). It appears that the GCF of these terms is 2. So, let's factor it out.

Step 2: Factor out the GCF

Now, we need to factor out the identified GCF (which is 2) from each term in the given polynomial: $$ 2x^2+6x-20 = 2(x^2+3x-10) $$

Step 3: Factor the quadratic

We are left with the quadratic \(x^2 + 3x - 10\) inside the parentheses. To factor this quadratic, let's look for two numbers that multiply to -10 and add up to 3. The numbers 5 and -2 fit these criteria. Thus, we can rewrite the quadratic as: $$ x^2+3x-10 = (x+5)(x-2) $$

Step 4: Write the fully factored polynomial

Finally, we combine the GCF and the factored quadratic to get the fully factored polynomial: $$ 2x^2+6x-20 = 2(x+5)(x-2) $$

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is crucial in simplifying polynomials by factoring. Essentially, the GCF of a set of terms is the largest factor that divides all of them without leaving a remainder.

To find the GCF, follow these steps:

• Identify the coefficients and constants in the polynomial's terms.
• Determine the highest number that can evenly divide each coefficient or constant.

For the polynomial given, the terms are \(2x^2\), \(6x\), and \(-20\). Each term needs the highest number that can divide them wholly. Here, the number 2 is the largest factor dividing each.

Factoring out the GCF results in a simplified polynomial, making further operations easier. In our example, factoring 2 out of \(2x^2 + 6x - 20\) gives \(2(x^2 + 3x - 10)\).

Quadratic Factoring

Quadratic factoring simplifies expressions of the form \(ax^2 + bx + c\). It involves rewriting a quadratic into a product of two binomials. This process is essential in solving polynomial equations.

We need two numbers in factoring quadratics that satisfy two conditions:

• Multiply to produce \(c\), the constant term.
• Add up to \(b\), the coefficient of the linear term.

For the quadratic \(x^2 + 3x - 10\), the task was to find two numbers that multiply to -10 and sum to 3. The numbers 5 and -2 work, allowing the quadratic to transform into \((x+5)(x-2)\).

Mastery of these techniques is vital to handle polynomials, enabling you to work with more complex algebraic expressions.

Polynomial Expression

A polynomial expression consists of multiple terms added together, involving variables raised to different powers. Each term is a product of a coefficient and a variable raised to an exponent.

Understanding polynomial expressions involves recognizing patterns and using operations that simplify or solve equations.

Consider the expression \(2x^2 + 6x - 20\). It's a polynomial with three terms, each comprising variable \(x\) with varying exponents. Dissecting such expressions into their parts allows easier manipulation.

Factoring polynomials, like in the given example, relies on breaking down complex expressions into simpler, more manageable pieces. Such skill involves identifying the GCF and applying methods like quadratic factoring. These methods transform polynomials into factored forms, providing a foundation for solving or simplifying them in future problems.