Problem 149 - Elementary Algebra

Question

Factor completely. \(5 x^{4}+10 x^{3}-75 x^{2}\)

Step-by-Step Solution

Answer

\(5x^2(x + 5)(x - 3)\)

Step 1 - Identify Common Factors

First, look for the greatest common factor (GCF) of all the terms in the polynomial. The GCF of 5, 10, and -75 is 5. Additionally, each term contains at least an \(x^2\). Therefore, the GCF is \(5x^2\). Factor out \(5x^2\) from each term.

Step 2 - Factor Out the Common Factor

Factor \(5x^2\) out of the polynomial: \(5x^4 + 10x^3 - 75x^2 = 5x^2(x^2 + 2x - 15)\)

Step 3 - Factor the Quadratic Expression

Next, factor the quadratic expression inside the parentheses: \(x^2 + 2x - 15\). Find two numbers that multiply to -15 and add to 2. These numbers are 5 and -3. Therefore, we factor the quadratic as: \(x^2 + 2x - 15 = (x + 5)(x - 3)\)

Step 4 - Combine All Factors

Substitute the factored quadratic expression back into the equation: \(5x^2(x^2 + 2x - 15) = 5x^2(x + 5)(x - 3)\)

Key Concepts

greatest common factor

When tackling any polynomial expression, the first step is to identify the Greatest Common Factor (GCF). The GCF is the highest factor that each term in the polynomial shares. For example, in the polynomial given: \(5x^4 + 10x^3 - 75x^2\), the coefficients are 5, 10, and -75. The GCF of these numbers is 5. Additionally, since each term includes at least \(x^2\), the highest power of x common to all terms is \(x^2\). So, the GCF for this polynomial is \(5x^2\). By factoring out \(5x^2\), we simplify the polynomial to facilitate further factorization. Hence, factoring out \(5x^2\) from our original polynomial, we get: \(5x^4 + 10x^3 - 75x^2 = 5x^2(x^2 + 2x - 15)\).

quadratic factorization

Quadratic factorization involves breaking down a quadratic expression into the product of two binomials. Consider the quadratic expression from the previous step: \(x^2 + 2x - 15\). To factorize this, we need to find two numbers that multiply to the constant term (-15) and add up to the coefficient of the linear term (2). These two numbers are 5 and -3 because: \(5 \times -3 = -15\) \(5 + (-3) = 2\). Thus, the quadratic \(x^2 + 2x - 15\) can be factored into the product of two binomials as follows: \(x^2 + 2x - 15 = (x + 5)(x - 3)\). This simpler form allows us to combine all factors together for the final expression.

polynomial expressions

Polynomial expressions consist of variables and coefficients structured into terms connected by addition, subtraction, or multiplication. Each term must adhere to the basic rules of algebra, involving variable powers and their coefficients. For instance, in \(5x^4 + 10x^3 - 75x^2\), there are three terms with degrees 4, 3, and 2 respectively. Working with polynomial expressions involves identifying patterns, common factors, and relevant algebraic techniques to simplify or factorize the expression. Polynomials can range from simple linear expressions to more complex higher-degree forms, requiring varying approaches to solving them effectively.

algebraic techniques

Algebraic techniques are essential tools for manipulating and solving expressions and equations. The following techniques help simplify polynomial expressions so we can factorize and solve them:

• Identifying and factoring out the Greatest Common Factor (GCF)
• Recognizing and using the Distributive Property
• Applying the difference of squares method
• Factoring quadratics into binomials

These techniques require practice and understanding of the fundamental properties of numbers and algebra. Utilizing these methods on polynomial expressions makes complex algebra manageable and solvable step-by-step. For example, in factorizing \(5x^4 + 10x^3 - 75x^2\), we first factor out the GCF, then factorize the resulting quadratic expression to finally combine all factors into a simplified product.