Problem 81 - Elementary Algebra

Question

Use any of the factoring methods to factor. Identify any prime polynomials. $$ 14 x^{2}+7 x-49 $$

Step-by-Step Solution

Answer

Factored form: 7(x - 1)(2x + 7)

Step 1: Identify the Coefficients

Identify the coefficients of the quadratic polynomial. Here, the polynomial is given as 14x^2 + 7x - 49 The coefficients are: a = 14, b = 7, c = -49.

Step 2: Factor Out the Greatest Common Factor (GCF)

Find and factor out the greatest common factor (GCF) from all terms. For the polynomial 14x^2 + 7x - 49, the GCF is 7. Factor out 7: 14x^2 + 7x - 49 = 7(2x^2 + x - 7)

Step 3: Factor the Remaining Quadratic Polynomial

Now focus on factoring the quadratic polynomial inside the parentheses: 2x^2 + x - 7 To factor this, look for two numbers that multiply to (a * c) = (2 * -7) = -14 and add to b = 1. These numbers are 7 and -2. Rewrite the middle term using these numbers: 2x^2 + 7x - 2x - 7

Step 4: Group and Factor by Grouping

Group the terms into pairs and factor out the common factors from each pair: (2x^2 + 7x) + (-2x - 7) Factor out the common factors from each group: x(2x + 7) - 1(2x + 7)

Step 5: Factor Out the Common Binomial

Now, factor out the common binomial factor (2x + 7): x(2x + 7) - 1(2x + 7) = (x - 1)(2x + 7)

Step 6: Combine the Factors

Combine the factor from Step 2 and the result from Step 5. Thus, the factored form of the original polynomial is: 14x^2 + 7x - 49 = 7(x - 1)(2x + 7)

Step 7: Identify Prime Polynomials

Since the polynomial is factored completely, there are no prime polynomials in this case.

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree 2. It has the general form ewlineewline ax^2 + bx + c. Here, a, b, and c are coefficients, and x is the variable. In our exercise, the quadratic polynomial given is 14x^2 + 7x - 49. The coefficient a is 14, b is 7, and c is -49. Quadratic polynomials are special because they form a parabola when graphed.
To factor a quadratic polynomial, we look for ways to express it as a product of two binomials, like (mx + n)(px + q). This often involves finding two numbers that multiply to give the product of the a and c terms and add to give the b term. In our case, we found these numbers to be 7 and -2, as they multiply to -14 (2 * -7) and add to 1 (7 + -2).

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number that divides all terms of a polynomial without leaving a remainder. It simplifies the polynomial by reducing it. For the polynomial 14x^2 + 7x - 49, the GCF is 7.
We factor out the GCF by dividing each term by 7, resulting in: ewline ewline 14x^2 + 7x - 49 = 7(2x^2 + x - 7).
By doing this, we simplify the polynomial and make further factoring steps easier. The whole essence of finding the GCF is to make the polynomial simpler and more factorable, helping us to break it down step by step.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four or more terms. It involves grouping terms with common factors and then factoring out those common factors.
In this exercise, after factoring out the GCF, we were left with 2x^2 + x - 7. To factor this, we rewrite the middle term x using our identified numbers 7 and -2: 2x^2 + 7x - 2x - 7.
Next, we group the terms: (2x^2 + 7x) + (-2x - 7).
We then factor out the common factors from each pair: ewlineewline x(2x + 7) - 1(2x + 7).
The term (2x + 7) is a common binomial factor, allowing us to factor it out, resulting in: (x - 1)(2x + 7).Grouping helps by breaking down the polynomial, making it straightforward to factor complex expressions.

Prime Polynomial

A prime polynomial is a polynomial that cannot be factored into the product of two or more non-constant polynomials. Essentially, it's like a prime number, which cannot be divided by anything other than 1 and itself.
In our exercise, after factoring the quadratic polynomial completely, we get the result 7(x - 1)(2x + 7). Since it can be factored fully, it means our polynomial is not a prime polynomial.
To identify a prime polynomial, attempt to factor it. If you cannot, even after several methods like grouping or special factorization techniques, then it is prime. However, with our exercise, we have successfully factored it, proving it's not prime.