Problem 40 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ 2 x^{2}-24 x+70 $$

Step-by-Step Solution

Answer

The trinomial \(2x^2 - 24x + 70\) factors to \(2(x - 5)(x - 7)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, we need to factor out the greatest common factor (GCF) from the given trinomial. In this case, each term in the trinomial (\(2x^2\), \(-24x\), and \(70\)) can be divided evenly by 2, so the GCF is 2.

Step 2: Factor Out the GCF

Divide each term in the trinomial by 2, and then rewrite the trinomial with the GCF factored out: \[ 2(x^2 - 12x + 35) \]

Step 3: Identify a Pair of Numbers for Factoring the Quadratic

Now, focus on the quadratic \(x^2 - 12x + 35\). We need to find two numbers that multiply to 35 (the constant term) and add up to -12 (the coefficient of the linear term). The numbers that satisfy this are -5 and -7.

Step 4: Factor the Quadratic

Use the numbers identified in the previous step to factor the quadratic: \[ x^2 - 12x + 35 = (x - 5)(x - 7) \]

Step 5: Write the Fully Factored Form

Combine the GCF with the factored quadratic to express the original trinomial in fully factored form: \[ 2(x - 5)(x - 7) \]

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is the largest number that divides all the terms of a polynomial without leaving a remainder. It's an essential first step when factoring trinomials because it simplifies the expression, making further factoring easier. For the polynomial \(2x^2 - 24x + 70\), each term shares a factor of 2.
- The term \(2x^2\) is divisible by 2 because 2 times \(x^2\) equals \(2x^2\). - For \(-24x\), dividing by 2 results in \(-12x\). - Finally, 70 divided by 2 is 35.
Factoring out the GCF is like peeling away the outer layer of a convoluted problem, revealing a simpler core trinomial that is easier to work with. Once we identify and factor out the GCF, we rewrite the expression like this: \[ 2(x^2 - 12x + 35) \]. This factored expression is simpler to handle for quadratic factoring.

Quadratic Factoring

Quadratic factoring involves finding two binomials whose product gives back the quadratic trinomial. The simplified trinomial from our previous step is \(x^2 - 12x + 35\).
To factor it, we need two numbers that multiply to the last term, 35, and add up to the middle coefficient, which is -12. This is often referred to as finding the magic pair.
- Here, the numbers \(-5\) and \(-7\) work since \(-5 \times -7 = 35\) and \(-5 + -7 = -12\).
These numbers split the middle term into two parts that help refactor the quadratic. Recognize that \(x^2 - 12x + 35\) will become:\((x-5)(x-7)\).This step must be handled with care and precision to ensure the factors correctly restore the quadratic when multiplied. It's akin to solving a puzzle, where each correct piece fits perfectly with the others.

Algebraic Expressions

An algebraic expression is made up of terms separated by plus or minus signs. Each term can be a number, a variable, or numbers and variables multiplied together. The given expression is \(2x^2 - 24x + 70\), which is a trinomial because it has three terms.
Understanding these expressions involves recognizing:

• Numbers (like 70 or -24), called constants.
• Variables (like \(x\)) that are placeholders for unknown values.
• exponents, where the power shows how many times the variable is used as a factor (\(x^2\) means \(x \times x\)).

Algebraic expressions follow rules that govern arithmetic operations, and understanding these rules aids in factoring. In this problem, we use the distributive property, combine like terms, and apply factorization techniques to completely break down or rebuild the algebraic structure. Being comfortable with these expressions is foundational for effectively handling more complex algebraic operations.