Problem 67 - Algebra 1: Concepts and Skills

Question

Which of the following is a correct factorization of \(72 x^{2}-24 x+2 ?\) F. \(-9(3 x-1)^{2}\) G. \(2(6 x-1)^{2}\) H. \(8(3 x-1)^{2}\) J. \(9(3 x-1)^{2}\)

Step-by-Step Solution

Answer

The correct factorization of the given equation \(72x^{2}-24x +2\) is \(2(6x-1)^2\) (Option G).

Step 1: Test Option F

Expand the factor \(-9(3x-1)^2\). The resulted equation will be \(-9 \cdot (9x^2 -6x +1) =-81x^2 +54x -9\), which does not match the given equation \(72x^2 -24x +2\). Therefore, this option is incorrect.

Step 2: Test Option G

Expand the factor \(2(6x-1)^{2}\). The resulted equation will be \(2 \cdot (36x^2 - 12x +1) =72x^{2}-24x +2\), which matches the given equation. Therefore, this option is correct.

Step 3: Test Remaining Options

Though we have found our answer in Option G, for learning's sake, we can verify that the remaining options H and J will not yield the correct factorization of the given equation when expanded.

Key Concepts

Polynomial Factorization

Factorization is a crucial skill in algebra that involves breaking down complex expressions into simpler, multipliable factors. Specifically, polynomial factorization refers to expressing a polynomial as a product of its factors. These factors are often polynomials of lower degrees, which, when multiplied together, give back the original polynomial.

For instance, let's consider the polynomial in the exercise, which is a quadratic polynomial given as \(72 x^{2}-24 x+2\). The task here is to determine which of the multiple-choice options correctly represents its factorization. In searching for the correct factorization, you would look for binomial expressions—expressions with two terms—that, when squared, would produce the original quadratic polynomial.

In our practice problem, after testing different options, the correct factorization was identified as \(2(6x-1)^{2}\). This illustrates how polynomial factorization can simplify expressions and make them more manageable for further mathematical operations, such as solving equations or graphing.

Quadratic Equations

Quadratic equations are a type of polynomial equation that specifically have the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). In the context of our exercise, the quadratic equation is not set to zero but is rather presented in a standard format ready for factorization.

Finding solutions to a quadratic equation typically involves factorization, completing the square, or using the quadratic formula. Factorization is advantageous when the equation is factorable into two binomial expressions, such as \(2(6x-1)^2\). This skill allows us to solve for the variable \(x\) by setting each factor equal to zero and solving for the roots of the equation. It is an elegant way to unravel the solutions of quadratic equations that students often encounter in algebra.

Solving Algebraic Equations

Solving algebraic equations, whether they are linear, quadratic, or polynomial, is a fundamental aspect of algebra. The process involves finding the value(s) of the variable(s) that make the equation true. With quadratic equations, as discussed above, this can involve factoring. However, the principles of solving algebraic equations extend to equations of all forms, requiring manipulation and simplification to isolate the variable.

Students should be equipped with a variety of techniques, such as factoring, using inverse operations, and applying properties of equality. For the problem at hand, after determining the correct factorization of the polynomial, one could proceed by setting each factor equal to zero, if the original problem involved solving for \(x\). By mastering these methods, algebra becomes a versatile tool for addressing a wide array of problems in mathematics.