To factor the polynomial [tex]\( 4a^2 - 21a + 5 \)[/tex], let's follow these steps:
1. Identify the polynomial structure:
The polynomial [tex]\( 4a^2 - 21a + 5 \)[/tex] is a quadratic expression in the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = -21 \)[/tex], and [tex]\( c = 5 \)[/tex].
2. Look for two numbers that multiply to [tex]\( ac \)[/tex] and add up to [tex]\( b \)[/tex]:
- Here, [tex]\( a \cdot c = 4 \times 5 = 20 \)[/tex].
- We need to find two numbers that multiply to 20 and add up to [tex]\( -21 \)[/tex].
3. Find such pair of numbers:
- The numbers [tex]\( -1 \)[/tex] and [tex]\( -20 \)[/tex] work because:
\begin{align}
-1 \times -20 & = 20 \quad (\text{product is } 20) \\
-1 + (-20) & = -21 \quad (\text{sums up to } -21)
\end{align}
4. Rewrite the middle term using these two numbers:
- Rewrite [tex]\( -21a \)[/tex] as [tex]\( -1a - 20a \)[/tex]:
[tex]\[
4a^2 - 21a + 5 = 4a^2 - 1a - 20a + 5
\][/tex]
5. Group the terms to factor by grouping:
- Group the terms:
[tex]\[
(4a^2 - 1a) + (-20a + 5)
\][/tex]
6. Factor out the greatest common factor (GCF) from each group:
- From the first group [tex]\( 4a^2 - 1a \)[/tex], factor out [tex]\( a \)[/tex]:
[tex]\[
a(4a - 1)
\][/tex]
- From the second group [tex]\( -20a + 5 \)[/tex], factor out [tex]\( -5 \)[/tex]:
[tex]\[
-5(4a - 1)
\][/tex]
7. Factor out the common binomial factor:
- Now, we have:
[tex]\[
a(4a - 1) - 5(4a - 1)
\][/tex]
- Notice that [tex]\( (4a - 1) \)[/tex] is a common factor. Factor this out:
[tex]\[
(a - 5)(4a - 1)
\][/tex]
So, the polynomial [tex]\( 4a^2 - 21a + 5 \)[/tex] factors to [tex]\( (a - 5)(4a - 1) \)[/tex].
