Question

3
factor the expression.
16x² + 8x
a -4x(4x + 2)
b 4(4x + 2)
c 4x(4x + 2)
d 4x(4x - 2)

Explanation:

Step1: Find the GCF of terms

The terms are \(16x^2\) and \(8x\). The GCF of 16 and 8 is 8, and the GCF of \(x^2\) and \(x\) is \(x\). So the GCF is \(8x\)? Wait, no, wait: 16x² and 8x. Let's factor each term:

16x² = 8x * 2x

8x = 8x * 1

Wait, no, 16x² = 4x 4x, 8x = 4x 2. Wait, maybe I made a mistake. Let's do it properly.

Factor out the greatest common factor (GCF) from \(16x^2 + 8x\).

First, find GCF of coefficients 16 and 8: GCF(16,8)=8.

GCF of variables: \(x^2\) and \(x\) is \(x\) (since \(x\) is the lowest power of x present).

So GCF is \(8x\)? Wait, no: 16x² = 8x 2x, 8x = 8x 1. So factoring out 8x: 8x(2x + 1). But that's not one of the options. Wait, maybe I made a mistake. Wait the options: let's check each option.

Option A: -4x(4x + 2) = -16x² -8x. Not matching.

Option B: 4(4x² + 2x) = 16x² +8x. Wait, no, the original is 16x² +8x. Wait, 4(4x² + 2x) = 16x² +8x. But the options have C: 4x(4x + 2). Let's expand C: 4x4x + 4x2 = 16x² +8x. Oh! Wait, I messed up the GCF. Let's re-express:

16x² +8x = 4x4x + 4x2 = 4x(4x + 2). Yes! Because 4x4x=16x², 4x2=8x. So that's correct.

Wait, why did I think GCF was 8x? Because 16 and 8 have GCF 8, but 16x² can be written as 4x4x, 8x as 4x2. So the GCF here is 4x, not 8x. Because 4x is a common factor. Let's check:

16x² ÷ 4x = 4x

8x ÷ 4x = 2

So factoring out 4x: 4x(4x + 2). Which is option C.

Let's verify each option:

A: -4x(4x + 2) = -16x² -8x. Not equal to 16x² +8x. Eliminate A.

B: 4(4x² + 2x) = 16x² +8x. Wait, that's also equal. But wait, the original expression is 16x² +8x. So B is 4(4x² +2x) = 16x² +8x. C is 4x(4x +2) = 16x² +8x. Wait, but which is correct? Wait, the problem is to factor the expression. Let's see: 16x² +8x. Let's factor step by step.

16x² +8x = 8x(2x +1) or 4x(4x +2) or 4(4x² +2x). But among the options, C is 4x(4x +2), which when expanded is 16x² +8x. Let's check:

4x4x = 16x², 4x2 = 8x. So yes, 4x(4x +2) = 16x² +8x. So that's correct.

So the correct option is C.

Step1: Identify GCF of terms

For \(16x^2\) and \(8x\), factor each:
\(16x^2 = 4x \cdot 4x\), \(8x = 4x \cdot 2\).
The common factor is \(4x\).

Step2: Factor out GCF

Factor \(4x\) from \(16x^2 + 8x\):
\(16x^2 + 8x = 4x(4x + 2)\).

Step3: Verify by expansion

Expand \(4x(4x + 2)\):
\(4x \cdot 4x + 4x \cdot 2 = 16x^2 + 8x\), which matches the original expression.

Answer:

C. \(4x(4x + 2)\)