Problem 86 - Beginning and Intermediate Algebra

Question

Factor completely by first taking out a negative common factor. $$-12 s^{4} t^{2}-4 s^{3} t^{3}+40 s^{2} t^{4}$$

Step-by-Step Solution

Answer

The short answer is: \(-4s^2t^2(s + 2t)(3s - 5t)\)

Step 1: Identify the greatest common factor (GCF)

The terms in the polynomial are: -12s^4t^2, -4s^3t^3, 40s^2t^4. To find the GCF, we'll examine the factors of each term, keeping in mind that we need to take out a negative common factor: - Numbers: -12 has factors of 1, 2, 3, 4, 6, and 12; -4 has factors of 1, 2, and 4; 40 has factors of 1, 2, 4, 5, 10, 20, and 40. The GCF for the coefficients is -4. - s variables: s^4, s^3, and s^2 share s^2 as the greatest common factor. - t variables: t^2, t^3, and t^4 share t^2 as the greatest common factor. So, the GCF that we need to take out as a negative common factor is: -4s^2t^2.

Step 2: Factor out -4s^2t^2 from the polynomial

Now we'll factor the GCF out of each term: $$ -4s^2t^2(\frac{-12s^4t^2}{-4s^2t^2} + \frac{-4s^3t^3}{-4s^2t^2} + \frac{40s^2t^4}{-4s^2t^2}) $$

Step 3: Simplify each term

Simplify each term inside the parentheses: $$-4s^2t^2(+3s^2 + s t - 10t^2) $$

Step 4: Factor the polynomial inside the parentheses (if possible)

We can factor the polynomial \(3s^2 + st - 10t^2\) further. We will use the AC method: Multiply the leading coefficient "3" by the constant "-10" to get "A * C = -30". Now, we'll look for two numbers that multiply to -30 and add up to "1" which is "st"s coefficient. $$3s^2 + st - 10t^2$$ The two numbers are 6 and -5. Rewrite the middle term as the sum of these two numbers times the corresponding variables. $$3s^2 + 6st - 5st - 10t^2$$ Now, factor by grouping: Group the first two terms and the last two terms: $$3s(s + 2t) - 5t(s + 2t)$$ Factor out the common binomial \((s+2t)\) from both terms: $$(s + 2t)(3s - 5t)$$

Step 5: Write the final factored polynomial

Now that we have factored the polynomial, we can put it all together: $$-4s^2t^2(s + 2t)(3s - 5t)$$

Key Concepts

greatest common factor

When factoring polynomials, identifying the greatest common factor (GCF) is the initial crucial step. The GCF is the largest factor that can evenly divide each term of the polynomial. To find it, you must examine each component of the terms: the numerical coefficients, and the powers of the variables.
For example, consider the polynomial \(-12s^4t^2 - 4s^3t^3 + 40s^2t^4\). Let’s start by identifying the GCF of the numerical coefficients: -12, -4, and 40.

• The factors for -12 are 1, 2, 3, 4, 6, 12.
• The factors for -4 are 1, 2, 4.
• The factors for 40 are 1, 2, 4, 5, 10, 20, 40.

The common factor among these numbers is -4, which is the greatest when considering a negative factor, suitable for our negative terms in the polynomial.
Next, note the variable factors:

• For variables, the smallest power of each variable common in all terms is taken.
• The **s** terms are \(s^4\), \(s^3\), and \(s^2\), with \(s^2\) being the smallest power shared.
• For the **t** terms, \(t^2\), \(t^3\), and \(t^4\), again \(t^2\) is the greatest power shared.

Thus, the greatest common factor for the polynomial is \(-4s^2t^2\), which you factor out from the polynomial to simplify each term.

AC method

The AC method is a beloved technique when it comes to factoring trinomials of the form \(ax^2 + bx + c\). This method is particularly effective for those cases when the leading coefficient \(a\), is not equal to 1.
In the original problem, we arrived at a trinomial \(3s^2 + st - 10t^2\) after factoring out the GCF. Let's break down the AC method further:

• First, multiply the leading coefficient \(a = 3\) by the constant term \(c = -10\), giving you \(A\times C = -30\).
• Now, identify two numbers that multiply to \(-30\) and add up to the middle coefficient \(b = 1\). These numbers are \(6\) and \(-5\).

The next step is to rewrite the middle term of the trinomial \(st\) as the sum of the two numbers found; hence, the expression becomes \(3s^2 + 6st - 5st - 10t^2\). This rearrangement allows you to then approach the problem using another method – factoring by grouping.

factoring by grouping

Factoring by grouping is a powerful technique especially when you have four terms. It allows you to factor a polynomial expression by grouping its terms into pairs, each of which has a common factor.
For the polynomial \(3s^2 + 6st - 5st - 10t^2\), this approach can be effectively applied:

• Identify and group the first two terms and the last two terms: \((3s^2 + 6st) - (5st + 10t^2)\).
• The objective is to factor each group separately by finding the GCF of each pair.
• The first pair \(3s^2 + 6st\) factors to \(3s(s + 2t)\).
• The second pair \(-5st - 10t^2\) factors to \(-5t(s + 2t)\).

Notice that both terms contain the common binomial \((s+2t)\). This common factor signifies that you can factor the expression as \((s+2t)(3s-5t)\).
This method of grouping significantly simplifies complex polynomials and allows you to see the expression in a different, often more manageable form.