Problem 63 - Elementary Algebra

Question

Use any of the factoring methods to factor. Identify any prime polynomials. $$ 18 x^{2}+5 x-7 $$

Step-by-Step Solution

Answer

The factors are \( (2x - 1) \) and \( (9x + 7) \).

Step 1 - Identify the coefficients

Identify the coefficients of the quadratic polynomial. Here they are: \( a = 18 \), \( b = 5 \), and \( c = -7 \).

Step 2 - Multiply the leading coefficient and the constant term

Calculate the product of the leading coefficient \( a \) and the constant term \( c \). \( 18 \times (-7) = -126 \).

Step 3 - Find two numbers that multiply to \(-126\) and add to \(5\)

Find two numbers that multiply to \(-126\) and add up to \(5\). These numbers are \( 14 \) and \(-9\) since \( 14 \times -9 = -126 \) and \( 14 + (-9) = 5 \).

Step 4 - Rewrite the middle term using the two numbers

Rewrite the quadratic polynomial using the numbers found: \( 18x^2 + 14x - 9x - 7 \).

Step 5 - Factor by grouping

Group the terms and factor them in pairs: \( (18x^2 + 14x) + (-9x - 7) \). Factor out the common terms: \( 2x(9x + 7) - 1(9x + 7) \).

Step 6 - Factor out the common binomial

Now factor out the common binomial factor: \( (2x -1)(9x + 7) \).

Step 7 - Identify if the polynomial is prime

Since the polynomial was successfully factored, it is not a prime polynomial. The factors are \( (2x - 1) \) and \( (9x + 7) \).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, which means they have the form: \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients, and \( x \) represents the variable. Quadratic equations are notable for their parabolic graph shapes
When solving these, one of the common methods is factoring.
Factoring involves expressing the quadratic polynomial as a product of two binomials. If a quadratic expression can be factored, it simplifies solving the equation for real roots.

Factoring by Grouping

Factoring by grouping is a method that simplifies quadratic equations. This technique involves these main steps:

• First, multiply the leading coefficient (the number in front of \( x^2 \)) by the constant term (the standalone number).
• Find two numbers that multiply to this product and add up to the middle coefficient (the number in front of \( x \)).
• Rewrite the quadratic expression by splitting the middle term into two terms that use the numbers found in the previous step.
• Group the terms in pairs.
• Factor out the common factor from each pair.
• Factor out the common binomial factor.

This process effectively breaks down the quadratic equation, making it easier to find its solution.

Coefficients in Polynomials

In polynomials, coefficients are the numerical factors that multiply the variable terms. For example, in the polynomial \( 18x^2 + 5x -7 \), the coefficients are:

• \( a = 18 \) for the \( x^2 \) term.
• \( b = 5 \) for the \( x \) term.
• \( c = -7 \) for the constant term.

Recognizing and working with these coefficients is crucial in solving quadratic equations because they determine the shape and position of the graph, and they are used in various factoring and solving methods.

Prime Polynomials

Prime polynomials are polynomials that cannot be factored into the product of two non-constant polynomials with integer coefficients. A quadratic polynomial \( ax^2 + bx + c \) could be considered prime if:

• No pair of integers exists that both multiply to \( ac \) and add up to \( b \).
• The polynomial cannot be simplified further.

In our example, we found the factors of the polynomial \( 18x^2 + 5x - 7 \), which are \( (2x - 1)(9x + 7) \). Since it can be factored into two binomials, it is not a prime polynomial.