Problem 69 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(p^{2}+11 p+30\)

Step-by-Step Solution

Answer

(p+5)(p+6)

Step 1 - Identify coefficients

Identify the coefficients from the trinomial of the form \(x^{2}+bx+c\). In the given trinomial \(p^{2}+11p+30\), the coefficient of \(p\) is \(b = 11\), and the constant term is \(c = 30\).

Step 2 - Find two numbers that multiply to 30 and add to 11

Look for two numbers that multiply to \(30\) (the constant term \(c\)) and add up to \(11\) (the coefficient \(b\)). These two numbers are \(5\) and \(6\) because \(5 \times 30=30\) and \(5 + 6 = 11\).

Step 3 - Write the factors

Express the trinomial as \(p^{2}+11p+30 = (p+5)(p+6)\). These are the factorized forms of the given trinomial.

Key Concepts

Identifying Coefficients

In order to factor a trinomial, the first step is to identify the coefficients present in the given expression. Let's consider the example trinomial expression: \[p^{2} + 11p + 30\]. Here, the coefficients are:

• The coefficient of the leading term, \(p^{2}\), is implicitly \(1\) because there is no number written in front of the \(p^2\).
• The coefficient of the next term, \(p\), is \(11\).
• The constant term at the end is \(30\).

These coefficients are important as they guide the next steps in the factoring process.

Finding Factor Pairs

The next step is to find two numbers that will help us factor the trinomial. Specifically, we need to find two numbers that multiply to give \(30\) (the constant term) and add up to \(11\) (the coefficient of \(p\)). To tackle this, list the factor pairs of \(30\):

• \((1, 30)\),
• \((2, 15)\),
• \((3, 10)\),
• \((5, 6)\)

Among these pairs, the pair \(5\) and \(6\) meets both conditions. They multiply to \(30\) (since \(5 \times 6 = 30\)) and add up to \(11\) (since \(5 + 6 = 11\)). Finding these specific factor pairs is essential for factoring the trinomial correctly.

Factoring Process

With the appropriate factor pairs identified, we can now proceed to write the trinomial in its factored form. The original trinomial is \(p^{2} + 11p + 30\). We found the numbers \(5\) and \(6\) that met our criteria. This allows us to express the trinomial as follows: \[(p + 5)(p + 6)\]. Breaking it down:

• \((p + 5)\): This represents one binomial factor with one of our identified numbers.
• \((p + 6)\): This represents the other binomial factor with the other identified number.

So, the factored form of the given trinomial \(p^{2} + 11p + 30\) is \((p + 5)(p + 6)\). Factoring lets us simplify and solve polynomial expressions more efficiently.