Problem 4 - Elementary Algebra

Question

Factor \(-14 m^{4}+28 m^{2}-7 m\).

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial \(-14 m^4 + 28 m^2 - 7 m\) is \(-7m(2(m^2 - 1)^2)\).

Step 1: Find the greatest common divisor (GCD)

Identify the GCD of the given coefficients -14, 28, and -7. The GCD in this case is 7.

Step 2: Factor out the GCD from the given polynomial

Factor out the GCD found in step 1 from the given polynomial: \(-14 m^4 + 28 m^2 - 7 m = -7m(2m^4 - 4m^2 + 1)\)

Step 3: Identify the type of expression inside the parentheses

Observe that the expression inside the parentheses is a quadratic expression in the form of \(ax^4 + bx^2 + c\), where \(x=m^2\). We can rewrite the expression as: \(2x^2 - 4x + 1\), with \(x = m^2\)

Step 4: Factor the quadratic expression

Now, let us factor the quadratic expression \(2x^2 - 4x + 1\). You can either factor by grouping, completing the square, or by using the quadratic formula. In this case, factoring by completing the square is the best method to apply: \(2x^2 - 4x + 1 = 2(x^2 - 2x) + 1\) Next, we look for the value \(b\) such that \((x - b)^2\) inside the parentheses would complete the square: \(b = \frac{1}{2}\cdot(-2) = -1\) So, the completed square is given by: \(2(x - 1)^2\) Now replace \(x\) with \(m^2\): \(2(m^2 - 1)^2\)

Step 5: Write the final factored form

Combine the factored GCD from step 2 and the factored quadratic expression from step 4 to get the final factored form: \(-7m(2(m^2 - 1)^2)\) So, the factored form of the given polynomial \(-14 m^4 + 28 m^2 - 7 m\) is \(-7m(2(m^2 - 1)^2)\).

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial stepping stone in factoring polynomials. It simplifies algebraic expressions by identifying the largest factor common to all the terms of the polynomial. This makes it easier to work with the polynomial and can often reveal simpler expressions to factor further. In our exercise, we wanted to factor the polynomial \(-14 m^4+28 m^2-7 m\).

• Step 1 was to find the GCD of the coefficients \(-14\), \(28\), and \(-7\). By inspection, we can see that each of these numbers share a common factor of \(7\).
• We notice that \(7\) encompasses the greatest factors shared among all terms.

Once identified, this GCD is then factored out of the entire polynomial expression. Doing so is like taking out the largest piece each term has in common, making the polynomial simpler to handle in subsequent steps.

Quadratic Expressions

Quadratic expressions form the backbone for many types of polynomials, especially in the context of high school algebra. They typically take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our exercise, after factoring out the GCD, we were left with \(2x^2 - 4x + 1\), recognizing \(x\) as the substitution \(m^2\).

• This transformed our polynomial to a more familiar quadratic format.
• Quadratic expressions are often easier to manage than polynomial expressions of a higher degree.

This transformation is part of our strategic approach to simplify and find the factors of complex polynomial expressions. Recognizing these forms allows us the flexibility of using established methods for further factoring.

Completing the Square

Completing the square is a technique specifically used in algebra to factorize quadratic expressions. It involves reshaping the quadratic into a perfect square trinomial plus or minus a constant, making it easier to solve or factor.
In our task, we wanted to factor \(2x^2 - 4x + 1\) using this method:

• First, we modified \(x^2 - 2x\) into \(2(x^2 - 2x) + 1\).
• Next, we identify how to complete \((x+b)^2\) by using \(b = \frac{1}{2}\cdot(-2) = -1\).
• This leads us to utilize the identity \((x - 1)^2\).

Upon successfully completing the square, we derived \(2(m^2 - 1)^2\) by substituting back to our original variable \(m^2\). This helped us recognize the structure within a quadratic, which wasn't initially obvious in the higher-degree polynomial.

Factored Form

The factored form is a way of presenting polynomials such that each term is expressed as a product of simpler terms. This not only simplifies the expression but also makes it easier to solve for roots or intersections. In our example of \(-14 m^4+28 m^2-7 m\), we worked towards breaking down the original complex polynomial into a neat product of smaller pieces.

• By extracting the GCD \(-7m\), we simplified the rest of the polynomial.
• Then, applying methods like completing the square on the resulting quadratic expression allowed us to further break it down.

The final factored form, \(-7m(2(m^2 - 1)^2)\), shows this approach. This lineup of terms means the polynomial is of simplest form where no further factoring is necessary using elementary algebra. Factored forms are crucial for solving polynomial equations and analyzing behavior like intersections with axes.