Problem 110 - Elementary Algebra

Question

(a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Factor \(45 c^{3}+33 c^{2}-168 c\) completely. Incorrect Answer: $$ \begin{aligned} &45 c^{3}+33 c^{2}-168 c \\ &=3 c\left(15 c^{2}+11 c-56\right) \end{aligned} $$

Step-by-Step Solution

Answer

The mistake was stopping at factoring out 3c first. The complete factorization is \(3c (3c + 7)(5c - 8)\).

Step 1 - Identify the mistake

Observe the given incorrect answer: \[45 c^{3}+33 c^{2}-168 c \] Incorrectly factored as: \[3 c\big(15 c^{2}+11 c-56\big)\] The common factor has been identified as 3c, but this factorization is incomplete because the quadratic term inside the parentheses can be further factored.

Step 2 - Find the Greatest Common Factor (GCF)

Factor out the greatest common factor (GCF) from the polynomial terms: \[45 c^{3}+33 c^{2}-168 c \] Identify the GCF of the coefficients 45, 33, and -168, and the common variable factor \(c\). The GCF for the coefficients is 3 and the common c:\[= 3c(15 c^{2}+11 c-56) \]

Step 3 - Factor the quadratic expression

Now, factor the quadratic expression inside the parentheses: \[15 c^2 + 11 c - 56\] We need to find two numbers whose product is \(15 \times -56 = -840\) and whose sum is 11.The numbers 35 and -24 work since \(35 \times -24 = -840\) and \(35 + (-24) = 11\). Rewrite the quadratic expression as: \[15 c^2 + 35 c - 24 c - 56\] Group the terms: \[15 c^2 + 35 c - 24 c - 56 = 5 c(3 c + 7) - 8(3 c + 7)\] Notice that \((3c + 7)\) is a common factor:\[= (3 c + 7)(5 c - 8) \]

Step 4 - Combine factors

Combine all the factored parts:\[3 c (15 c^2 + 11 c - 56) = 3c (3c + 7)(5c - 8) \]Thus, the completely factored form is: \[= 3c (3c + 7)(5c - 8) \]

Key Concepts

Greatest Common Factor

Polynomials can be simplified by first finding the Greatest Common Factor (GCF). The GCF is the highest factor that divides all the terms in the polynomial, including the numerical coefficients and variable parts. For example, in the polynomial 45c^3 + 33c^2 - 168c, we need to find the highest number and the variable power that can divide each term.
- For the coefficients: 45, 33, and -168, the GCF is 3.
- The common variable factor overall is c.
Therefore, the GCF for the polynomial is 3c. Factoring the GCF out, we get:
3c(15c^2 + 11c - 56).
Recognizing and factoring out the GCF helps in simplifying the polynomial, making further factorization easier.

Quadratic Expression

A quadratic expression is any polynomial that can be written in the form
ax^2 + bx + c. Here, the degree of the polynomial is 2 (the highest exponent of the variable is 2). For the polynomial 15c^2 + 11c - 56, we see that it is a quadratic expression because the term with the highest power is 15c^2.
The goal with quadratic expressions is typically to factor them into a product of two binomials, when possible. This involves finding two numbers that multiply to the product of a and c (in this case, 15 and -56) but add up to b (here, 11).
In our example, the quadratic expression 15c^2 + 11c - 56 is factored into:
(3c + 7)(5c - 8).
Breaking the quadratic expression into simpler terms allows us to express the polynomial in its completely factored form.

Polynomial Factoring

Factoring polynomials involves breaking them down into simpler parts, known as factors, that can be multiplied to give the original polynomial. The process usually starts with finding the GCF of all terms and then factoring any remaining quadratic expressions. Here's a step-by-step approach using our example.
1. Identify and factor out the GCF, which is 3c in: 45c^3 + 33c^2 - 168c = 3c(15c^2 + 11c - 56).
2. Factor the quadratic expression 15c^2 + 11c - 56. We look for two numbers that multiply to -840 (15 * -56) and add to 11 (b). These numbers are 35 and -24.
3. Rewrite the quadratic expression: 15c^2 + 35c - 24c - 56.
4. Group the terms and factor each group: 5c(3c + 7) - 8(3c + 7).
5. Notice that (3c + 7) is a common factor, so factor it out: (3c + 7)(5c - 8).
6. Combine everything: 3c(3c + 7)(5c - 8).
Thus, the completely factored form of the polynomial is:
3c(3c + 7)(5c - 8).
Factoring polynomials simplifies them, making it easier to understand their behavior and solve equations where they appear.