Problem 53 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$2 a^{2}+7 a b+5 b^{2}$$

Step-by-Step Solution

Answer

The trinomial factors as \(2a^{2}+7ab+5b^{2} = (2a+5b)(a+b)\)

Step 1: Factorizing

To factorize, we need to find two numbers that add up to 7 and multiply to 10 (because our a=2 and c=5 hence a*c=10). These two numbers are 2 and 5. Now we rewrite the middle term (7ab) as the sum of (2ab and 5ab). Therefore we can rewrite our trinomial as : \(2a^{2}+2ab+5ab+5b^{2}\)

Step 2: Factor by grouping

We can now factor by grouping these terms: \(2a(a+b)+5b(a+b)\) Then, since the expression \(a+b\) is a common factor in both terms, we can factor it out: \((2a+5b)(a+b)\)

Step 3: Checking the factorization with FOIL

To confirm the factorization is correct, we can multiply the factors out using the FOIL method: \((2a+5b)(a+b) = 2a^{2}+ b(2a) + a(5b) + 5b^{2} = 2a^{2}+7ab+5b^{2}\) Thus, this confirms the factorization is correct

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases involving variables raised to various powers, often with coefficients and constants. In our exercise, the expression is a trinomial. A trinomial is a specific type of polynomial that has three terms, which in this case is represented by \(2a^2 + 7ab + 5b^2\). The components of a polynomial expression include:

• Coefficients: These are the numbers multiplying the variables, such as 2, 7, and 5 in our trinomial.
• Variables: The letters such as \(a\) and \(b\) in our expression, which can represent unknown values.
• Exponents: These indicate the power to which the variables are raised, such as 2 in \(a^2\).

Polynomial expressions can be categorized by degree, which is the highest power of the variable present in the expression. In the given trinomial, the degree is 2, making it a quadratic polynomial. Understanding these components helps in factoring, a key skill in simplifying and solving equations.

FOIL Method

The FOIL method is a technique used to multiply two binomials. The acronym "FOIL" stands for First, Outside, Inside, Last, which describes the order in which you multiply the terms of the binomials. Let's apply this to our exercise:

• First: Multiply the first terms of each binomial. For \((2a+5b)(a+b)\), this is \(2a \cdot a = 2a^2\).
• Outside: Multiply the outer terms. Here, that's \(2a \cdot b = 2ab\).
• Inside: Multiply the inner terms. So, \(5b \cdot a = 5ab\).
• Last: Multiply the last terms of each binomial, giving \(5b \cdot b = 5b^2\).

After applying FOIL, you combine like terms to simplify. You'll find the original trinomial \(2a^2 + 7ab + 5b^2\), confirming the factorization is correct.
The FOIL method is especially handy in checking factorization or expanding polynomials, ensuring your solution is accurate.

Algebraic Factorization

Algebraic factorization is the process of breaking down a complex polynomial into simpler expressions that can be multiplied together to yield the original polynomial. The given trinomial \(2a^2 + 7ab + 5b^2\) was factored by rewriting the middle term, using the method of grouping:

• Find numbers that multiply to the product of the first and last coefficients and add up to the middle coefficient.
• Rewrite the middle term as a sum of two terms using these numbers. Here, \(7ab\) is rewritten as \(2ab + 5ab\).
• Group terms that can be factored by pulling out common factors: \(2a(a+b) + 5b(a+b)\).
• Factor out the common expression, resulting in \((2a+5b)(a+b)\).

Factoring transforms a polynomial from a sum or difference of terms to a product of factors. This simplification aids in solving equations, understanding polynomial behavior, and in calculus for finding derivatives and integrals.