Problem 48 - Algebra 2

Question

Factor each expression. $$ 5 x^{2}+5 x-10 $$

Step-by-Step Solution

Answer

The factored form of the expression \(5x^{2}+5x-10\) is \(5(x+2)(x-1)\).

Step 1: Find the Greatest Common Factor (GCF)

The greatest common factor of \(5x^{2}\), \(5x\), and \(-10\) is \(5\). Divide each term of the expression by \(5\). This gives us: \(5(x^{2}+x-2)\)

Step 2: Factor the quadratic trinomial

Next, factor the quadratic trinomial \(x^{2}+x-2\). We find two numbers that multiply to \(-2\) (the constant term) and add to \(1\) (the coefficient of \(x\)). The numbers that satisfy this are \(2\) and \(-1\), thus the factorization is \((x+2)(x-1)\). Substitute that back in to our expression from Step 1 to get \(5(x+2)(x-1)\)

Step 3: Check the solution

We can check the solution by multiplying \(5\), \((x+2)\), and \((x-1)\) together to ensure it equals our original expression. When we do this, we confirm that \(5(x+2)(x-1)\) simplifies to \(5x^{2}+5x-10\), our original expression, thus confirming that the solution is correct.

Key Concepts

Greatest Common Factor

When factoring polynomials, the first step is often to identify the Greatest Common Factor (GCF). The GCF is the largest number or expression that can evenly divide each term of the polynomial. This step simplifies the polynomial and makes further factoring easier.

To find the GCF of the expression \(5x^2 + 5x - 10\):

• Identify the coefficients and constant: \(5\), \(5\), and \(-10\). These share a common factor of \(5\).
• Extract \(5\) out of each term: \(5(x^2 + x - 2)\).

By dividing each term by \(5\), the polynomial becomes simpler, setting the stage for further factorization.

Quadratic Trinomial

A quadratic trinomial is a polynomial of degree 2 that has three terms. It typically takes the form \(ax^2 + bx + c\). To factor a quadratic trinomial like \(x^2 + x - 2\), we look for two numbers that both multiply to the constant term \(c\) and add to the linear coefficient \(b\).

In the trinomial \(x^2 + x - 2\):

• The constant term is \(-2\), and we need factors of \(-2\).
• The linear coefficient is \(1\), so our two numbers must add up to \(1\).

Upon analyzing, we find the numbers \(2\) and \(-1\) meet these criteria because \(2 \times -1 = -2\) and \(2 + -1 = 1\). Hence, the trinomial factors as \((x + 2)(x - 1)\).

This factorization simplifies the expression and allows us to solve or manipulate the original polynomial more easily.

Binomial Factorization

Once we have decomposed the quadratic trinomial into binomials, the polynomial expression becomes a product of these binomials. This process of converting a polynomial into the product of smaller polynomials is known as binomial factorization.

Applying binomial factorization to the expression \(5(x+2)(x-1)\), we have:

• The expression inside the parentheses, \(x+2\) and \(x-1\), represent the factorized form of \(x^2 + x - 2\).
• This makes the entire expression look like \(5 \times (x+2) \times (x-1)\).

Understanding binomial factorization is crucial as it allows us to focus on how expressions can be broken down into simpler parts, which are easier to handle in algebraic operations.