Problem 100 - Algebra A Combined Function

Question

Find all positive values of \(c\) so that each trinomial is factorable. A student in your class factored \(6 x^{2}+7 x+1\) as \((3 x+1)(2 x+1)\). Write down how you would explain the student's error.

Step-by-Step Solution

Answer

The student incorrectly expanded the product as \((3x+1)(2x+1)\) which gives \(6x^2 + 5x + 1\), not \(6x^2 + 7x + 1\). The correct factorization is \((6x + 1)(x + 1)\).

Step 1: Understand the Problem

We want to factor the trinomial \(6x^2 + 7x + 1.\) The student tried to factor it as \((3x+1)(2x+1)\). We need to verify if this factorization is correct or find the correct factorization.

Step 2: Verify the Student's Factorization

Expand the expression \((3x+1)(2x+1)\) to verify if it's correct:\[(3x+1)(2x+1) = 3x(2x) + 3x(1) + 1(2x) + 1(1) = 6x^2 + 3x + 2x + 1 = 6x^2 + 5x + 1.\]The obtained expression \(6x^2 + 5x + 1\) is not equal to the original trinomial \(6x^2 + 7x + 1\). Therefore, the student's factorization is incorrect.

Step 3: Find the Correct Factorization

To factor \(6x^2 + 7x + 1\), start by identifying two numbers whose product is \(6 imes 1 = 6\) and whose sum is 7. These two numbers are 6 and 1.

Step 4: Rewrite the Middle Term

Rewrite the middle term \(7x\) as \(6x + x\):\[6x^2 + 7x + 1 = 6x^2 + 6x + x + 1.\]

Step 5: Factor by Grouping

Group the terms:\[(6x^2 + 6x) + (x + 1).\]Factor out the common factors:\[= 6x(x + 1) + 1(x + 1).(6x + 1)(x + 1).\]

Step 6: Verify the Correct Factorization

Expand \((6x + 1)(x + 1)\) to ensure it matches the original trinomial:\[(6x + 1)(x + 1) = 6x^2 + 6x + x + 1 = 6x^2 + 7x + 1.\] The factorization is correct.

Key Concepts

Trinomial Factorization

When we look at trinomials, like the expression \(6x^2 + 7x + 1\), the goal is to break them down into a product of two binomials. This process is called trinomial factorization and can be incredibly useful in simplifying algebraic expressions.

A trinomial is a type of algebraic expression that consists of three terms. In the trinomial \(6x^2 + 7x + 1\), we are dealing with a quadratic trinomial due to the highest power being \(x^2\). For successful factorization, we'll often find pairs of numbers that will allow us to rewrite the middle term (\(7x\) in this case) and systematically group the terms.

The challenge lies in picking the correct pairs of numbers whose product and sum help factor the expression correctly. This often involves a process of trial and error, checking if our chosen factors multiply back to give the original trinomial. For \(6x^2 + 7x + 1\), the correct factorization turns out to be \((6x + 1)(x + 1)\), verified through polynomial expansion.

Algebraic Expressions

Algebraic expressions are composed of variables and constants combined using mathematical operations. The expression \(6x^2 + 7x + 1\) is an example of a polynomial, more specifically, a trinomial. Understanding algebraic expressions is crucial because it enables the solving of equations, factorization, and the simplification of complex expressions.

Key components of such expressions include:

• **Terms**: Parts of the expression separated by plus or minus signs.
• **Coefficients**: Numbers that multiply the variables (e.g., 6 in \(6x^2\)).
• **Variables**: Symbols that stand in for unknown values, like \(x\).
• **Constants**: Numbers without variables, like 1.

Understanding these components helps in dealing with more complex algebraic processes like factorization, where the aim is to break down the polynomial into simpler elements.

Polynomial Expansion

Polynomial expansion is the reverse process of factorization. It involves multiplying two expressions to restore the original polynomial. This mathematical technique checks whether a proposed factorization is accurate. In other words, by expanding a factorized form, you determine whether it equals the original expression.

For example, to check the factorization of \(6x^2 + 7x + 1\) as \((6x + 1)(x + 1)\), we use the distributive property:

• Multiply \((6x + 1)\) by \(x\) to get \(6x^2 + x\).
• Multiply \((6x + 1)\) by \(1\) to get \(6x + 1\).

Adding these results gives us the original expression, \(6x^2 + 7x + 1\). This confirms that the factorization is correct.

Polynomial expansion thus ensures every factorization step is accurate, providing confidence in solving algebraic problems.