Problem 83 - Elementary Algebra

Question

Use any of the factoring methods to factor. Identify any prime polynomials. $$ 4 c^{2}+8 c-5 $$

Step-by-Step Solution

Answer

(2c + 5)(2c - 1)

Step 1: Identify the Form

Recognize that the given polynomial is a quadratic in the form of \(ax^2 + bx + c\) where \(a = 4\), \(b = 8\) and \(c = -5\).

Step 2: Multiply \(a\) and \(c\)

Multiply the coefficient of the quadratic term \(a\) with the constant term \(c\). Here it is \(4 \times (-5) = -20\).

Step 3: Find Two Numbers

Find two numbers that multiply to \(-20\) and add to \(b = 8\). The numbers \(10\) and \(-2\) work because \(10 \times (-2) = -20\) and \(10 + (-2) = 8\).

Step 4: Rewrite the Middle Term

Rewrite the polynomial by splitting the middle term, \(8c\), using the two numbers found: \(4c^2 + 10c - 2c - 5\).

Step 5: Factor by Grouping

Group the terms into pairs: \((4c^2 + 10c) - (2c + 5)\). Factor out the common factors from each pair: \(2c(2c + 5) - 1(2c + 5)\).

Step 6: Factor the Common Binomial

Factor out the common binomial factor \((2c + 5)\): \((2c + 5)(2c - 1)\).

Step 7: Conclusion

The given quadratic polynomial \(4c^2 + 8c - 5\) is factored as \((2c + 5)(2c - 1)\). This is not a prime polynomial since it can be factored.

Key Concepts

quadratic polynomials

Quadratic polynomials are expressions of the form \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. These polynomials have a highest degree of 2, meaning the largest exponent of the variable is 2. Quadratic polynomials are often written in standard form, which is useful for both solving and factoring. For example, the polynomial \(4c^2 + 8c - 5\) is in standard form with \(a = 4\), \(b = 8\), and \(c = -5\). Understanding the structure of quadratic polynomials is crucial for applying factoring methods correctly. The main goal is to transform the polynomial into a product of simpler polynomials.

factoring methods

There are several methods to factor quadratic polynomials, including:

• Trial and Error: Looking for pairs of factors of the quadratic term and constant term that add up to the middle coefficient.
• Grouping Method: Splitting the middle term to present the polynomial as a product of binomials.
• Difference of Squares: Used for polynomials in the form \(a^2 - b^2\) which factors into \((a + b)(a - b)\).

The choice of method can depend on the specific polynomial and its coefficients. In our example, we used the grouping method.

prime polynomials

Prime polynomials are polynomials that cannot be factored into simpler polynomials with integer coefficients. They are analogous to prime numbers in arithmetic. When factoring, always check for prime polynomials to avoid unnecessary steps. For instance, if you attempt to factor a quadratic polynomial and find no valid factorizations, you might conclude it is prime. In this case, our given polynomial \(4c^2 + 8c - 5\) was successfully factored into \((2c + 5)(2c - 1)\), showing it is not a prime polynomial.

grouping method

The grouping method for factoring polynomials is particularly effective for quadratic equations. Here’s a step-by-step guide:
1. Multiply \(a\) and \(c\): This gets us a product that aids in breaking the middle term.
2. Find Two Numbers: Look for two numbers that multiply to \(ac\) and add to \(b\).
3. Rewrite the Polynomial: Split the middle term using these two numbers.
4. Group the Terms: Pair the terms to allow factoring by grouping.
5. Factor Each Group: Take out the common factors from each group.
6. Combine: The common binomial factor should appear, allowing further factoring. In our problem, we rewrote the polynomial as \(4c^2 + 10c - 2c - 5\) and grouped them into \((4c^2 + 10c) - (2c + 5)\).

polynomial multiplication

Understanding polynomial multiplication helps in both expanding and factoring expressions. Multiplying polynomials involves distributing each term in one polynomial to every term in the other. For quadratic polynomials, you can use the FOIL method:

• First: Multiply the first terms of each polynomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms.

When multiplying \((2c + 5)(2c - 1)\), you'll get \(2c \times 2c = 4c^2\), \(2c \times -1 = -2c\), \(5 \times 2c = 10c\), and \(5 \times -1 = -5\). Combining these gives back the original polynomial: \(4c^2 + 8c - 5\). This reverse process helps verify the correctness of factoring.