Problem 65 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(m^{2}+12 m+11\)

Step-by-Step Solution

Answer

(m + 1)(m + 11)

Step 1 - Identify coefficients

Identify the coefficients in the trinomial. Here, the equation is given as: In this case, the coefficients are:

Step 2 - Find factor pairs of the constant term

Find pairs of factors of the constant term (11) that add up to the coefficient of the middle term. Here, we need to find factors of 11 that add up to 12.

Step 3 - Identify the factors

List the factor pairs of 11: - (1, 11). Check if their sum is equal to 12: 1+11 = 12

Step 4 - Write factors

Since the factors are correctly identified as 1 and 11, the trinomial will be split accordingly: (m + 1)(m + 11).

Step 5 - Verify

Multiply the factors to check the accuracy: (m + 1)(m + 11) = m^2 + 11m + m + 11 = m^2 + 12m + 11. The product matches the original trinomial.

Key Concepts

Coefficients

When factoring trinomials, the first step is to identify the coefficients. Coefficients are the numerical values that multiply the variables in a polynomial. For the given trinomial, which is \(m^2 + 12m + 11\), the coefficients are:

• 1 (coefficient of \(m^2\))
• 12 (coefficient of \(m\))
• 11 (constant term)

. Recognizing these numbers helps frame the equation for the next steps.

Factor Pairs

In the second step, we focus on the factor pairs of the constant term. Factor pairs are groups of two numbers that, when multiplied together, give the original number. For our trinomial \(m^2 + 12m + 11\), we focus on the constant term, which is 11.
Factor pairs of 11 are:

• (1, 11)

. We then check if the sum of these factor pairs equals the coefficient of the middle term. In this case:
\(1 + 11 = 12\). Since it matches, we use these in the next step to factor the expression.

Constant Term

Understanding the constant term is crucial in the factoring process. It is the last number in the polynomial that does not have a variable. For \(m^2 + 12m + 11\), the constant term is 11. In problem-solving, we need to find numbers that multiply to give us this constant term while also considering their relationship with the other coefficients. Identifying correct factor pairs of the constant term aligns the equation for successful factoring.

Middle Term

The middle term in a trinomial is the term with a single variable multiplied by its coefficient. In \(m^2 + 12m + 11\), the middle term is 12m. The goal of finding factor pairs was to match their sum to this middle term.
We checked:

• Factor pairs of 11 which are (1, 11).

Their sum is 12, matching the middle term.
This allows us to split and rewrite the polynomial as \((m + 1)(m + 11)\).