Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomial. \(7 x^{2}+48 x-7\)

Step-by-Step Solution

Answer

The polynomial is factored as \((x + 7)(7x - 1)\).

Step 1: Identify the Coefficients

The polynomial given is in the form of a quadratic equation, which is generally written as \(ax^2 + bx + c\). Here, \(a = 7\), \(b = 48\), and \(c = -7\).

Step 2: Find Two Numbers That Multiply to ac

The product of \(a\) and \(c\) is \(7 \times (-7) = -49\). We need two numbers that multiply to \(-49\) and add up to \(48\).

Step 3: Determine Suitable Numbers

The suitable numbers are \(49\) and \(-1\) because \(49 \times (-1) = -49\) and \(49 + (-1) = 48\).

Step 4: Rewrite the Middle Term

Rewrite the middle term, \(48x\), using the numbers found: \(7x^2 + 49x - 1x - 7\).

Step 5: Factor by Grouping

Group the terms: \((7x^2 + 49x) + (-1x - 7)\). Factor each group: \(7x(x + 7) - 1(x + 7)\).

Step 6: Factor the Common Binomial

The common binomial is \(x + 7\), so factor it out: \((x + 7)(7x - 1)\).

Step 7: Verify the Result

Expand \((x + 7)(7x - 1)\) to check: \((x)(7x) - (x)(1) + (7)(7x) - (7)(1) = 7x^2 + 49x - x - 7\), which simplifies to \(7x^2 + 48x - 7\). The factorization is correct.

Key Concepts

Quadratic Equation

In mathematics, a quadratic equation is any equation that can be rearranged in standard form as \( ax^2 + bx + c = 0 \), where \( x \) represents an unknown variable, and \( a, b, \) and \( c \) are constants with \( a eq 0 \). Quadratic equations are essential for understanding polynomial expressions and factorization. Quadratics form parabolas when graphed on a coordinate plane and have distinctive properties such as roots or solutions, which can be real or complex numbers. Remember, when you come across an equation like \( 7x^2 + 48x - 7 \), identifying it as quadratic helps you apply the correct methods for solving, such as factorization or using the quadratic formula.

Coefficients

Coefficients are the numbers that multiply the variables or powers in a polynomial. They play a crucial role in defining the polynomial equation. For the quadratic equation in the problem \( 7x^2 + 48x - 7 \), we identify the coefficients as follows:

• \( a = 7 \) (coefficient of \( x^2 \))
• \( b = 48 \) (coefficient of \( x \))
• \( c = -7 \) (constant term)

The coefficients influence the shape and position of the graph of a polynomial. They are also used in various techniques, such as factoring, to simplify and solve polynomial equations. Recognizing the coefficients, you can determine the necessary steps for factorization and solve the quadratic effectively.

Factoring by Grouping

Factoring by grouping is a method used to simplify polynomials when they can be broken into groups that share a common factor. In the equation \( 7x^2 + 48x - 7 \), after identifying that two numbers, \( 49 \) and \(-1\), needed to multiply to \(-49\) and add to \(48\), we can break down the polynomial:

• Rewrite \( 48x \) as \( 49x - 1x \).
• Group the terms: \( (7x^2 + 49x) \) and \( (-1x - 7) \).
• Factor each group: \( 7x(x + 7) - 1(x + 7) \).
• Notice the common factor \( (x + 7) \).

This method of grouping is especially helpful when a straightforward solution is not visible. By factoring common elements in groups, we can simplify the process and factor the entire polynomial.

Binomial

A binomial is a polynomial expression that contains exactly two terms. In the factorization process of the given polynomial \( 7x^2 + 48x - 7 \), the resulting expression is \((x + 7)(7x - 1)\) after factoring by grouping. Both \(x + 7\) and \(7x - 1\) are binomials.

• \(x + 7\): the first binomial
• \(7x - 1\): the second binomial

Binomials can be easily multiplied out to verify that they recombine to form the original polynomial. Understanding binomials helps in both the multiplication and factorization of higher degree polynomials, proving to be a fundamental component in algebraic manipulation.

Verify Factorization

Verifying factorization involves expanding the factors to ensure they equal the original polynomial. In this case, we took the factors \((x + 7)(7x - 1)\) and expanded them as follows:

• \(x \times 7x = 7x^2 \)
• \(x \times -1 = -x \)
• \(7 \times 7x = 49x \)
• \(7 \times -1 = -7 \)

Combine these results to get \(7x^2 + 49x - x - 7\), simplifying to the original polynomial \(7x^2 + 48x - 7\). This step is crucial because it confirms the accuracy of your factoring process, ensuring no mistakes were made during the operation. Double-checking your work through expansion helps in solidifying your understanding and confidence in algebraic procedures.