Problem 69 - Algebra and Trigonometry

Question

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 Common Factors

Look at each term in the expression and determine if there is a common factor. The terms are \(x^4\), \(2x^3\), and \(-3x^2\). Each term contains a factor of \(x^2\). So, the common factor is \(x^2\).

Step 2: Factor Out the Common Factor

Factor \(x^2\) out of each term:\[x^2(x^2) + x^2(2x) - x^2(3)\]This simplifies to:\[x^2(x^2 + 2x - 3)\]

Step 3: Factor the Quadratic Expression

Now, we need to factor the quadratic expression \(x^2 + 2x - 3\). To factor this, we need two numbers that multiply to \(-3\) and add to \(2\). These numbers are \(3\) and \(-1\).

Step 4: Write the Factored Form

The quadratic expression \(x^2 + 2x - 3\) can be factored into the product of two binomials, \((x + 3)(x - 1)\). Thus, the fully factored form of the original expression is:\[x^2(x + 3)(x - 1)\]

Step 5: Check Your Work

To ensure the factoring is correct, multiply the factors back out to see if you get the original expression:\[x^2(x + 3)(x - 1) = x^2(x^2 + 2x - 3) = x^4 + 2x^3 - 3x^2\]Thus, the original expression matches.

Key Concepts

Common Factor

In mathematics, finding a common factor is often the first step in simplifying algebraic expressions. It's a bit like spotting a pattern or a repeated part in different terms. When you scan an expression, like the one given in the problem, you look for anything that each term shares. Think about it as searching for a "least common denominator", but here it's about factors instead.

In the expression \(x^4 + 2x^3 - 3x^2\), each term features the variable \(x\). But not just any \(x\)— an \(x^2\) is what they all have in common:

• First term: \(x^4 = x^2 \times x^2\)
• Second term: \(2x^3 = 2 \times x^2 \times x\)
• Third term: \(-3x^2 = -3 \times x^2\)

Recognizing \(x^2\) as the common factor, we can "factor it out". It's akin to factoring numbers where you take common numbers out of a series of terms to simplify them. What's left after factoring this out will guide you to further simplification.

Quadratic Expression

Once the common factor \(x^2\) is taken out from the original expression, we're left with a simpler quadratic expression: \(x^2 + 2x - 3\). Quadratic expressions are polynomials of degree two, typically written as \(ax^2 + bx + c\). Here, our quadratic already has those components arranged neatly:

• \(a = 1\)
• \(b = 2\)
• \(c = -3\)

The goal here is to factor this expression further. Think of it as breaking a number into its prime components. For quadratic expressions, factoring involves finding two numbers that not only multiply to give the constant term \(-3\) (the product \(a \times c\)) but also add up to the middle term's coefficient \(2\). These numbers—\(3\) and \(-1\)—make it possible to rewrite the quadratic as the product of two simpler expressions.

Binomial Factors

Binomial factors are simpler expressions usually of the form \(px + q\), which come into play when you break down a quadratic expression. With the quadratic \(x^2 + 2x - 3\), we identify the numbers \(3\) and \(-1\) that satisfy our factor conditions—the perfect duo to split the middle term into two parts:

• \(x^2 + 2x - 3 = (x + 3)(x - 1)\)

Each of these binomials, \((x + 3)\) and \((x - 1)\), represents a factor of the quadratic. Individually, they can't describe the entire expression, but together, they tell the complete story. By multiplying them, you can recreate the original quadratic part of the equation.

Understanding binomial factors helps to simplify complex expressions and solve equations. It's like discovering smaller pieces that fit together to form a larger whole, allowing us to see the expression's structure more clearly.