Question

4
describe the process of factoring trinomials when $a = 1$ and when $a \
eq 1$
provide an example of each. be sure to factor each example.

Explanation:

For \(a=1\) (trinomial form \(x^2+bx+c\)):

1. Find two integers \(m\) and \(n\) such that \(m+n = b\) and \(m \times n = c\).
2. Rewrite the trinomial as \((x+m)(x+n)\).

Example: Factor \(x^2+5x+6\)

• Find \(m=2\), \(n=3\) (since \(2+3=5\), \(2 \times 3=6\))
• Result: \((x+2)(x+3)\)

For \(a

eq 1\) (trinomial form \(ax^2+bx+c\)):

1. Calculate \(a \times c\), then find two integers \(m\) and \(n\) where \(m+n = b\) and \(m \times n = a \times c\).
2. Split the middle term \(bx\) into \(mx+nx\), then factor by grouping.

Example: Factor \(2x^2+7x+3\)

• Calculate \(2 \times 3=6\); find \(m=6\), \(n=1\) (since \(6+1=7\), \(6 \times 1=6\))
• Rewrite: \(2x^2+6x+x+3\)
• Group and factor: \(2x(x+3)+1(x+3)=(2x+1)(x+3)\)

Answer:

Process for \(a=1\):

1. Find \(m,n\): \(m+n=b\), \(m \times n=c\)
2. Factor as \((x+m)(x+n)\)

Example: \(x^2+5x+6=(x+2)(x+3)\)

Process for \(a

eq 1\):

1. Find \(m,n\): \(m+n=b\), \(m \times n=a \times c\)
2. Split middle term, factor by grouping

Example: \(2x^2+7x+3=(2x+1)(x+3)\)