Problem 76 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(q^{2}-13 q+36\)

Step-by-Step Solution

Answer

(q - 9)(q - 4)

Step 1 - Identify coefficients

Identify the coefficients in the trinomial. Here, the trinomial is of the form \(x^{2}+bx+c\). For the given trinomial \(q^{2}-13q+36\), \(b = -13\) and \(c = 36\).

Step 2 - Find two numbers that multiply to c and add to b

Find two numbers that multiply to \(c = 36\) and add up to \(b = -13\). These two numbers are \(-9\) and \(-4\) because \((-9) \times (-4) = 36\) and \((-9) + (-4) = -13\).

Step 3 - Write the trinomial as a product of binomials

Using the numbers found in Step 2, write the trinomial as a product of two binomials: \(q^{2}-13q+36 = (q - 9)(q - 4)\).

Key Concepts

Coefficients in Trinomials

Coefficients in a trinomial are the numbers in front of the variables. In the given trinomial, which is of the form \(x^{2} + bx + c\), we identify each coefficient. The coefficient in front of \(x^{2}\) is usually 1 (unless stated otherwise), making it the 'leading coefficient.' The coefficient \(b\) is in front of the linear term \(x\), and \(c\) is the constant term.
In our specific example, \(q^{2} - 13q + 36\), the coefficient \(b\) is \(-13\) and the constant term \(c\) is 36. Understanding these coefficients is crucial because they guide the factorization process. Remember, factorization is the reverse process of expanding an expression. By identifying coefficients correctly, you make sure to set the foundation for the multiplication and grouping steps.

Multiplication of Numbers

A key part in factoring trinomials is finding two numbers that multiply to the constant term \(c\) and add up to the linear coefficient \(b\). In this case, we need two numbers that multiply to 36 (our constant term, \(c\)) and add up to -13 (our linear coefficient, \(b\)).
The multiplication step is about exploring all the factor pairs of 36. Factor pairs of a number are pairs of integers whose product is the number itself. For 36, these pairs are:

• (1, 36)
• (2, 18)
• (3, 12)
• (4, 9)
• (6, 6)

After finding these pairs, the next step is to check which pair, when added together, gives you the linear coefficient \(b\). For 36, suitable pairs come from recognizing that we need negative terms for the sum to be negative. The pair that works here is \-9\ and \-4\ because:

• \(-9 \times -4 = 36\)
• \(-9 + -4 = -13\)

This multiplication and addition check is crucial for correctly breaking down the trinomial into its factors.

Binomials

A binomial is a polynomial with exactly two terms. In the context of factoring trinomials, once we identify the two numbers that work (from our multiplication step), we then express the trinomial as a product of two binomials.
For our example \(q^{2} - 13q + 36\), once we identified \-9\ and \-4\, we can write our trinomial as a product of two binomials:
\((q - 9)(q - 4)\)
This works because if we were to expand these binomials using the distributive property (also known as FOIL method for First, Outer, Inner, Last), we'd get back to the original trinomial:
\((q - 9)(q - 4) = q^{2} - 4q - 9q + 36 = q^{2} - 13q + 36\)
In summary, the binomials derived from the factors we found will multiply together to re-form the original trinomial. This closing checks your work and confirms correct factorization.