What is the factorization of [tex]3x^2 - 8x + 5[/tex]?

A. [tex](3x - 1)(x + 5)[/tex]
B. [tex](3x + 5)(x - 1)[/tex]
C. [tex](3x - 1)(x - 5)[/tex]
D. [tex](3x - 5)(x - 1)[/tex]



Answer :

To factorize the quadratic expression [tex]\(3x^2 - 8x + 5\)[/tex], we need to find two binomials whose product gives us the original expression.

Given a quadratic expression of the form [tex]\(ax^2 + bx + c\)[/tex]:
[tex]\[ 3x^2 - 8x + 5 \][/tex]

1. First, identify the coefficients: [tex]\(a = 3\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = 5\)[/tex].

2. We need to find two numbers that multiply to [tex]\(a \cdot c = 3 \cdot 5 = 15\)[/tex] and add up to [tex]\(b = -8\)[/tex].

3. The pairs of numbers that multiply to 15 are:
[tex]\[ (1, 15), (-1, -15), (3, 5), (-3, -5) \][/tex]

4. Among these pairs, the pair that adds up to -8 is [tex]\((-3, -5)\)[/tex].

5. Now rewrite the middle term [tex]\(-8x\)[/tex] using these numbers:
[tex]\[ 3x^2 - 3x - 5x + 5 \][/tex]

6. Group the terms in pairs and factor out the common factors in each pair:
[tex]\[ 3x(x - 1) - 5(x - 1) \][/tex]

7. Now, factor out the common binomial [tex]\((x - 1)\)[/tex]:
[tex]\[ (x - 1)(3x - 5) \][/tex]

Thus, the correct factorization of the quadratic expression [tex]\(3x^2 - 8x + 5\)[/tex] is:
[tex]\[ (x - 1)(3x - 5) \][/tex]

So, among the given options, the correct choice is:

[tex]\[(3x - 5)(x - 1)\][/tex]