Problem 59 - Algebra 1: Concepts and Skills

Question

Completely factor the expression. $$ 15 x^{4}-50 x^{3}-40 x^{2} $$

Step-by-Step Solution

Answer

The completely factorized expression is \(5x^{2}(3x + 2)(x - 4)\).

Step 1: Identify the common factor

Look at each term in the equation \(15x^{4} - 50x^{3} - 40x^{2}\) and identify the common factor. In this case, the common factor is \(5x^{2}\) because it is the highest degree and value that can divide into each term.

Step 2: Factor out the common factor

Factor out \(5x^{2}\) from each term in the equation. This gives \(5x^{2}(3x^{2} - 10x - 8)\) after dividing each term of the equation by \(5x^{2}\).

Step 3: Factorize the quadratic expression

Further factorize the expression within the parenthesis. This is a quadratic expression that can be factorized by searching for two numbers that multiply to \(3*-8 = -24\), and add to -10. Here those numbers are -12 and 2. Factorize to give \(5x^{2}(3x + 2)(x - 4)\).

Key Concepts

Common Factor

Understanding the concept of a 'common factor' is crucial when working with polynomial expressions. A common factor is a term that is shared among all terms in a polynomial, and it is found by identifying the greatest common divisor of the coefficients and the lowest power of any variables present. In the given exercise, the expression \(15x^{4} - 50x^{3} - 40x^{2}\) contains the common factor \(5x^{2}\). This is because 5 is a divisor of 15, 50, and 40, and \(x^{2}\) is the highest power of \(x\) that evenly divides into all three terms.

Factoring Quadratics

Factoring quadratics is a method used to break down quadratic expressions into two binomials. A quadratic expression is in the form \(ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \eq 0\). The goal is to find two binomials that when multiplied together give the original quadratic. To factor the quadratic part of the provided exercise, which is \(3x^{2} - 10x - 8\), we search for two numbers that multiply to \(a \times c\) and add up to \(b\). Here, those numbers are -12 and 2 because \(3 \times (-8) = -24\) and \(\left( -12 \right) + \left( 2 \right) = -10\). Consequently, the quadratic can be factored as \(3x + 2)(x - 4)\).

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables, coefficients, and exponents. The general form of a polynomial expression in a single variable \(x\) is \(a_{n}x^{n} + a_{n-1}x^{n-1} + \ldots + a_{2}x^{2} + a_{1}x + a_{0}\), where \(a_{n}\) through \(a_{0}\) are constants, and \(n\) is a non-negative integer. In the exercise, \(15x^{4} - 50x^{3} - 40x^{2}\) is a polynomial of degree 4 because the highest exponent of \(x\) is 4. Polynomials can be factored into products of polynomials of lower degree, as demonstrated by factoring out the common factor and then further factorizing the resulting quadratic expression.

Algebraic Techniques

Several algebraic techniques are used to manipulate and simplify expressions, with factoring being one of the most important. Factoring simplifies expressions and solves equations efficiently. Mastering factoring involves understanding methods like finding common factors, factoring quadratics, and using special products (such as the difference of squares or perfect square trinomials). After extracting the common factor \(5x^{2}\) from the polynomial expression, we apply the technique for factoring quadratics. The initial complex expression \(15x^{4} - 50x^{3} - 40x^{2}\) is thus simplified into \(5x^{2}(3x + 2)(x - 4)\), a product of simpler binomial expressions, showcasing effective algebraic manipulation.