To factor the quadratic polynomial [tex]\( 9a^2 - a - 8 \)[/tex], follow these steps:
1. Identify the Polynomial: Start with the quadratic polynomial [tex]\( 9a^2 - a - 8 \)[/tex].
2. Look for Two Binomials: We want to express [tex]\( 9a^2 - a - 8 \)[/tex] in the form [tex]\((pa + q)(ra + s)\)[/tex], where [tex]\( p, q, r, \)[/tex] and [tex]\( s \)[/tex] are numbers we need to find.
3. Determine Coefficients:
- The product of the first terms of the binomials should equal the coefficient of [tex]\( a^2 \)[/tex].
- The product of the constant terms should equal the constant term of the polynomial.
- The sum of the outer and inner products should equal the middle coefficient (the coefficient of the [tex]\( a \)[/tex] term).
Given that our polynomial is [tex]\( 9a^2 - a - 8 \)[/tex]:
- The coefficient of [tex]\( a^2 \)[/tex] is [tex]\( 9 \)[/tex].
- The constant term is [tex]\( -8 \)[/tex].
4. Find Factors:
- We search for pairs of numbers that multiply to [tex]\( 9 \times (-8) = -72 \)[/tex] (this is the product of the first and last coefficients).
- We need a pair that also adds up to the middle coefficient, which is [tex]\( -1 \)[/tex].
Evaluating possible pairs, we find that:
- [tex]\( 8 \times (-9) = -72 \)[/tex]
- And [tex]\( 8 + (-9) = -1 \)[/tex]
5. Rewrite the Middle Term:
Using the pair [tex]\( 8 \)[/tex] and [tex]\( -9 \)[/tex], we rewrite [tex]\( -a \)[/tex] as:
[tex]\[ 9a^2 - 9a + 8a - 8 \][/tex]
6. Factor by Grouping:
We group the terms:
[tex]\[ (9a^2 - 9a) + (8a - 8) \][/tex]
Factor common terms from each group:
[tex]\[ 9a(a - 1) + 8(a - 1) \][/tex]
Notice the common binomial factor [tex]\((a - 1)\)[/tex]:
[tex]\[ (9a + 8)(a - 1) \][/tex]
So, the factored form of [tex]\( 9a^2 - a - 8 \)[/tex] is:
[tex]\[ (a - 1)(9a + 8) \][/tex]
Answer: The factorization of [tex]\( 9a^2 - a - 8 \)[/tex] is [tex]\((a - 1)(9a + 8)\)[/tex].
