To factor the polynomial [tex]\( z^2 + 9z + 20 \)[/tex], we follow these steps:
1. Identify the General Form:
The polynomial is in the standard quadratic form [tex]\( az^2 + bz + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = 20 \)[/tex].
2. Set Up the Factoring Form:
Since the leading coefficient [tex]\( a = 1 \)[/tex], we are looking for two binomials of the form [tex]\( (z + m)(z + n) \)[/tex] such that when multiplied, they give the original polynomial [tex]\( z^2 + 9z + 20 \)[/tex].
3. Find the Coefficients [tex]\( m \)[/tex] and [tex]\( n \)[/tex]:
We need to find two numbers [tex]\( m \)[/tex] and [tex]\( n \)[/tex] such that:
- Their product is equal to the constant term [tex]\( c = 20 \)[/tex] (i.e., [tex]\( m \cdot n = 20 \)[/tex]).
- Their sum is equal to the linear coefficient [tex]\( b = 9 \)[/tex] (i.e., [tex]\( m + n = 9 \)[/tex]).
4. Solve for [tex]\( m \)[/tex] and [tex]\( n \)[/tex]:
Let’s look at pairs of factors of 20:
- [tex]\( 1 \cdot 20 \)[/tex]
- [tex]\( 2 \cdot 10 \)[/tex]
- [tex]\( 4 \cdot 5 \)[/tex]
Among these pairs, we find that [tex]\( 4 \cdot 5 = 20 \)[/tex] and [tex]\( 4 + 5 = 9 \)[/tex].
5. Write the Factored Form:
Therefore, the numbers [tex]\( m \)[/tex] and [tex]\( n \)[/tex] that satisfy the given conditions are 4 and 5. Hence, we can express the polynomial as:
[tex]\[
z^2 + 9z + 20 = (z + 4)(z + 5)
\][/tex]
So, the factored form of the polynomial [tex]\( z^2 + 9z + 20 \)[/tex] is:
[tex]\[
(z + 4)(z + 5)
\][/tex]
