Certainly! Let's factor the given polynomial expression step-by-step.
The polynomial we need to factor is:
[tex]\[ 5x^2 - 16x + 3 \][/tex]
### Step 1: Identify the coefficients from the polynomial.
The polynomial is in the standard form [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = -16\)[/tex]
- [tex]\(c = 3\)[/tex]
### Step 2: Find two numbers that multiply to [tex]\( ac \)[/tex] and add to [tex]\( b \)[/tex].
First, calculate [tex]\( ac \)[/tex]:
[tex]\[ ac = 5 \cdot 3 = 15 \][/tex]
Now, we need to find two numbers that multiply to 15 and add to [tex]\(-16\)[/tex].
These numbers are [tex]\(-15\)[/tex] and [tex]\(-1\)[/tex], because:
[tex]\[ -15 \cdot -1 = 15 \][/tex]
[tex]\[ -15 + (-1) = -16 \][/tex]
### Step 3: Rewrite the middle term using the two numbers.
Rewrite [tex]\(-16x\)[/tex] as [tex]\(-15x - x\)[/tex]:
[tex]\[ 5x^2 - 15x - x + 3 \][/tex]
### Step 4: Factor by grouping.
Group the terms into pairs:
[tex]\[ (5x^2 - 15x) + (-x + 3) \][/tex]
Factor out the greatest common factor (GCF) from each pair:
[tex]\[ 5x(x - 3) - 1(x - 3) \][/tex]
### Step 5: Factor out the common binomial factor.
Notice that [tex]\((x - 3)\)[/tex] is a common factor:
[tex]\[ (5x - 1)(x - 3) \][/tex]
So, the factored form of the polynomial is:
[tex]\[ \boxed{(5x - 1)(x - 3)} \][/tex]
This is the factorization of the polynomial [tex]\( 5x^2 - 16x + 3 \)[/tex].
