Problem 28 - Algebra 2 and Trigonometry

Question

In \(27-39,\) factor each polynomial completely. $$ 5 x^{2}-15 x+10 $$

Step-by-Step Solution

Answer

The fully factored form is \(5(x - 1)(x - 2)\).

Step 1: Identify the Greatest Common Factor (GCF)

Look at each term in the polynomial \(5x^2 - 15x + 10\) and determine the greatest common factor. Each term (\(5x^2\), \(-15x\), and \(10\)) is divisible by 5. So, the GCF is 5.

Step 2: Factor Out the GCF

Divide each term by the GCF (which is 5) and factor it out of the polynomial: \(5x^2 - 15x + 10 = 5(x^2 - 3x + 2)\).

Step 3: Factor the Quadratic Trinomial

Look at the quadratic trinomial \(x^2 - 3x + 2\). Find two numbers that multiply to 2 (the constant term) and add to -3 (the coefficient of the \(x\) term). These numbers are -1 and -2.

Step 4: Express the Trinomial as a Product of Binomials

Break down the trinomial using the numbers found in Step 3. Write the trinomial as \((x - 1)(x - 2)\).

Step 5: Write the Fully Factored Polynomial

Combine the factors: factor out the GCF and the factored trinomial. The complete factorization of \(5x^2 - 15x + 10\) is \(5(x - 1)(x - 2)\).

Key Concepts

greatest common factor

The Greatest Common Factor (GCF) is an essential tool when factoring polynomials, simplifying calculations, and solving complex equations. To identify the GCF, look for the largest number that can exactly divide each term of the polynomial without leaving a remainder.

For instance, in the polynomial \(5x^2 - 15x + 10\), examine each term: \(5x^2\), \(-15x\), and \(10\). You will notice that each term is divisible by 5. Therefore, the GCF is 5. Once the GCF is identified, factor it out of the polynomial. This involves dividing each term by 5 and simplifying, resulting in \(5(x^2 - 3x + 2)\).

Factoring out the GCF simplifies the polynomial, making further factorization easier. It’s like peeling off the outermost layer of complexity, allowing you to work with a simpler expression. Always take the time to identify and factor out the GCF as the initial step in polynomial factorization!

quadratic trinomials

Quadratic trinomials, specifically those of the form \(ax^2 + bx + c\), are a common sight in algebra. The quadratic trinomial in our example is \(x^2 - 3x + 2\). The task is to express it as a product of two binomials.

To do this, find two numbers that multiply to the constant term (2 in this case) and add up to the middle coefficient (-3 here). For \(x^2 - 3x + 2\), the numbers are \(-1\) and \(-2\) because:

• \(-1 \times -2 = 2\)
• \(-1 + -2 = -3\)

This process is like solving a puzzle where each number has to perfectly fit into both conditions. Understanding how to factor these trinomials is crucial because it reveals the roots of the equation, which help in graphing and solving algebraic problems.

factoring binomials

Factoring binomials involves breaking down an expression into two binomials. This is the final piece of the puzzle in the factorization process. Once you have identified the numbers that multiply and add to the necessary coefficients, rewrite the quadratic trinomial as the product of binomials.

For our expression \(x^2 - 3x + 2\), we found the numbers \(-1\) and \(-2\). This allows us to express the trinomial as \((x - 1)(x - 2)\). Here’s how it all comes together:

• The roots \(x - 1\) and \(x - 2\) correspond to the solutions of the equation \(x^2 - 3x + 2 = 0\).
• When solved, \(x = 1\) and \(x = 2\) are the points where the graph intersects the x-axis.

Factoring binomials not only facilitates finding these solutions but also aids in simplifying complex algebraic expressions, making them easier to work with in further calculations.