What are the factors of 3x2 + 23x − 8? (3x − 4)(x + 2) (3x − 2)(x + 4) (3x − 8)(x + 1) (3x − 1)(x + 8)

To factor the quadratic expression $3x^2 + 23x - 8$, we will follow the steps outlined.

1. Break Down the Problem

We need to factor the expression $3x^2 + 23x - 8$. This requires finding two numbers that multiply to give the product of the coefficient of $x^2$ (which is 3) and the constant term (which is -8), and add to give the coefficient of $x$ (which is 23).

2. Relevant Concepts

For a quadratic expression in the form $ax^2 + bx + c$, we look for factors of $a \cdot c$ that sum to $b$:

• $a = 3$
• $b = 23$
• $c = -8$

We need to find $ac = 3 \times -8 = -24$.

3. Analysis and Detail

We need two numbers that multiply to -24 and sum to 23.

The pairs of factors of -24 include:

• (1, -24)
• (-1, 24)
• (2, -12)
• (-2, 12)
• (3, -8)
• (-3, 8)
• (4, -6)
• (-4, 6)

Out of these, the pair that sums to 23 is $-1$ and $24$.

4. Verify and Summarize

Now, we can express the quadratic as: $3x^2 + 24x - x - 8$ Grouping terms: $(3x^2 + 24x) + (-x - 8)$ Factoring by grouping: $3x(x + 8) - 1(x + 8)$ Factoring out the common factor $(x + 8)$: $(3x - 1)(x + 8)$

Final Answer

The factors of $3x^2 + 23x - 8$ are $(3x - 1)(x + 8)$.