Problem 65 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ x^{4}+2 x^{3}-3 x^{2} $$

Step-by-Step Solution

Answer

The completely factored form is \(x^2(x + 3)(x - 1)\).

Step 1: Identify the Greatest Common Factor

First, we look for any common factors in all the terms of the expression. The expression is \(x^4 + 2x^3 - 3x^2\). The greatest common factor (GCF) of these terms is \(x^2\), as it is present in each term.

Step 2: Factor Out the Greatest Common Factor

We factor \(x^2\) out of each term in the expression. The expression becomes \(x^2(x^2 + 2x - 3)\).

Step 3: Factor the Quadratic Expression

Next, we factor the quadratic expression \(x^2 + 2x - 3\). We need two numbers that multiply to \(-3\) (the constant term) and add to \(2\) (the coefficient of the linear term). These numbers are \(3\) and \(-1\).

Step 4: Write the Factored Form

Using the numbers \(3\) and \(-1\) found in the previous step, we write the quadratic as \((x + 3)(x - 1)\). Therefore, the completely factored form of the expression is \(x^2(x + 3)(x - 1)\).

Key Concepts

Greatest Common Factor

When you are aiming to factor a polynomial, the first step is often to identify the Greatest Common Factor, or GCF. The GCF is the largest factor that divides all terms of the polynomial. This process simplifies the expression and makes further factoring steps easier.
For example, in the polynomial \(x^4 + 2x^3 - 3x^2\), each term includes the factor \(x^2\). So, \(x^2\) is the greatest common factor of this polynomial. By extracting this GCF, the polynomial becomes more manageable. Once you factor out \(x^2\), you simplify the expression to \(x^2(x^2 + 2x - 3)\).

Key steps to find the GCF include:

• List out the factors of each term.
• Identify the common factors.
• Select the greatest of these common factors.

Recognizing and factoring the GCF simplifies the expression and aids in further solving.

Quadratic Expression

Once you have simplified a polynomial by factoring out the GCF, you may be left with a quadratic expression to factor. Quadratic expressions typically have the form \(ax^2 + bx + c\). In our example, after finding the GCF, the quadratic expression left is \(x^2 + 2x - 3\).
Quadratics can often be factored by finding two numbers that multiply to give you the constant term \(c\) and add to the linear coefficient \(b\). For \(x^2 + 2x - 3\), we look for two numbers that:

• Multiply to \(-3\) (the constant term).
• Add to \(2\) (the linear coefficient).

These numbers are \(3\) and \(-1\). This approach allows you to break down the quadratic into linear factors, forming the expression \((x+3)(x-1)\). This process is crucial, as it reveals the roots of the quadratic equation.

Factoring Using Coefficients

Factoring quadratics using coefficients is a strategic method that involves the coefficients of the terms in the quadratic expression. In general, this process helps transform a quadratic into a product of two binomials. You specifically seek two numbers that satisfy the conditions based on the product and sum of certain coefficients.
For example, for the quadratic \(x^2 + 2x - 3\), the coefficients involved are:

• \(a = 1\)
• \(b = 2\)
• \(c = -3\)

The technique involves finding two numbers that multiply to the product of \(a\) and \(c\) (which is \(-3\)) and add up to \(b\) (which is \(2\)). By carefully considering these two numbers, \(3\) and \(-1\), divide \(b\) into these factors to help rewrite and simplify into two binomials.

Factoring using coefficients not only confirms the factorization results in a stepwise manner but also enhances the understanding of the relationships among the quadratic's components.