Problem 18 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$a^{2}+2 a b-63 b^{2}$$

Step-by-Step Solution

Answer

The polynomial is factored as \((a + 9b)(a - 7b)\).

Step 1: Identify the Pattern

The polynomial \(a^2 + 2ab - 63b^2\) is a quadratic trinomial in the variable \(a\). The general form for factoring such expressions is \(ax^2 + bx + c\). Here, \(a = 1\), \(b = 2b\), and \(c = -63b^2\).

Step 2: Look for Integer Factors

To factor the trinomial, find two integers whose product is the coefficient of \(c = -63b^2\) and sum is the middle term coefficient, which is \(2b\).

Step 3: List Possible Factor Pairs

The number \(-63\) can be factored as \(-1 \times 63\), \(-3 \times 21\), \(-7 \times 9\), and reversed pairs. You need the pair that adds up to the middle coefficient, \(2\).

Step 4: Choose Correct Pair and Factor

The pair \(9b\) and \(-7b\) satisfies the conditions, as \(9b + (-7b) = 2b\). Therefore, the factorization is: \( (a + 9b)(a - 7b) \).

Step 5: Verify the Factorization

Verify by expanding \((a + 9b)(a - 7b)\): \(a^2 - 7ab + 9ab - 63b^2 = a^2 + 2ab - 63b^2\). The verified factorization is correct.

Key Concepts

quadratic trinomials

Quadratic trinomials are polynomials of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. These are second-degree polynomials because the highest power of \( x \) is 2. They are called trinomials since they consist of three terms. Understanding how to recognize and factor these trinomials is crucial for simplifying expressions and solving equations.
One common method to factor quadratic trinomials is to look for two numbers that multiply together to give the product of \( a \times c \) and add up to \( b \). This method is often used when \( a = 1 \), simplifying the factorization process.

• Identifying the coefficients \( a \), \( b \), and \( c \) based on the general form.
• Using these coefficients to find a pair of numbers that satisfy the multiplication and addition conditions.
• Rewriting the middle term and factoring by grouping, if necessary.

By mastering these techniques, you can easily factor and work with quadratic trinomials.

integer factorization

Integer factorization is key when working with polynomials, particularly in finding factors that satisfy the quadratic trinomial's properties. The main task is to identify pairs of integers that multiply to a particular number, called the product.
For example, in the exercise, we were looking for two numbers that multiply to \(-63\), which is the product of the constant term and the coefficient of the squared term when factored by variables. More importantly, these integers also need to sum up to a specific middle coefficient, here \(2b\).

• The integers \(9b\) and \(-7b\) were found because they satisfy both \(-7b \times 9b = -63b^2\) and \(9b + (-7b) = 2b\).
• This dual-condition search is what makes integer factorization useful in identifying the correct binomial factors for quadratic trinomials.

Keep in mind that not all polynomials can be factored using integers. If no integer pairs meet the criteria, the polynomial might be deemed not factorable with integers.

polynomial verification

Polynomial verification involves confirming that the factorization obtained is correct. This is a vital step to ensure that the factors truly reproduce the original polynomial, closing the factoring process loop.
Verification is done by expanding the factored form back into its original polynomial and checking if each term matches. For instance, after factorizing our given quadratic trinomial as \((a + 9b)(a - 7b)\), verification by expansion produces:
\[ (a + 9b)(a - 7b) = a^2 - 7ab + 9ab - 63b^2 \]
Simplifying these terms gives back \(a^2 + 2ab - 63b^2\), which matches our initial polynomial. This confirms correct factorization.

• Always multiply each term and simplify.
• Recheck the polynomial after expansion to avoid errors.

This step assures you that the factors you've found are precise and reliable.