Question

factor out the greatest common factor. 32x⁴ + 24x³ - 16x² - 8x ?x(□x³ + □x² - 2x - 1)

Explanation:

Step1: Find GCF of coefficients and variables

First, find the GCF of the coefficients \(32, 24, -16, -8\). The GCF of these numbers is \(8\). For the variable part, the lowest power of \(x\) is \(x^1\) (i.e., \(x\)). So the GCF of the terms is \(8x\). Wait, but let's check the given form. The given form has a factor of \(x\) outside, so let's factor out \(x\) first, then find the GCF of the remaining coefficients.

After factoring out \(x\) from each term: \(32x^4 = x \cdot 32x^3\), \(24x^3 = x \cdot 24x^2\), \(-16x^2 = x \cdot (-16x)\), \(-8x = x \cdot (-8)\). Now we need to find the GCF of \(32, 24, -16, -8\). The GCF of \(32,24,16,8\) is \(8\). Wait, but in the given box, after factoring out \(x\), the first box is a number, then inside the parentheses, the first term is \(\square x^3\), second is \(\square x^2\), etc. Wait, let's re-examine the original expression: \(32x^4 + 24x^3 - 16x^2 - 8x\). Let's factor out \(8x\) first? Wait, no, the given form is \([?]x(\square x^3 + \square x^2 - 2x - 1)\). So let's factor out \(x\) first, then factor out the GCF of the remaining coefficients.

After factoring out \(x\), we get \(x(32x^3 + 24x^2 - 16x - 8)\). Now we need to factor out the GCF from \(32x^3 + 24x^2 - 16x - 8\). The GCF of \(32,24,16,8\) is \(8\)? Wait, no, because in the parentheses, the third term is \(-2x\), so let's see: \(32x^3 \div 16 = 2x^3\)? Wait, no, let's check the coefficients. Let's divide each coefficient by the first box's number. Wait, the given parentheses has \(-2x\) and \(-1\). Let's see:

Original expression: \(32x^4 + 24x^3 - 16x^2 - 8x\)

Factor out \(x\): \(x(32x^3 + 24x^2 - 16x - 8)\)

Now, let's factor out a number from \(32x^3 + 24x^2 - 16x - 8\) such that when we divide \(-16x\) by that number, we get \(-2x\). So \(-16x \div (\text{number}) = -2x\) implies that the number is \(\frac{-16x}{-2x} = 8\)? Wait, no, \(-16x \div 8 = -2x\), yes! And \(-8 \div 8 = -1\), which matches the last term. Then \(32x^3 \div 8 = 4x^3\)? Wait, no, wait the first term in the parentheses is \(\square x^3\), second is \(\square x^2\). Wait, no, let's do it step by step.

Wait, the given form is \([?]x(\square x^3 + \square x^2 - 2x - 1)\). Let's let the first box (the [?]) be \(a\), the second box (inside parentheses, first \(\square\)) be \(b\), third be \(c\). So:

\(a x (b x^3 + c x^2 - 2x - 1) = a b x^4 + a c x^3 - 2a x^2 - a x\)

This should equal \(32x^4 + 24x^3 - 16x^2 - 8x\). So we can set up equations:

1. Coefficient of \(x^4\): \(a b = 32\)
2. Coefficient of \(x^3\): \(a c = 24\)
3. Coefficient of \(x^2\): \(-2a = -16\)
4. Coefficient of \(x\): \(-a = -8\)

Wait, from the coefficient of \(x\): \(-a = -8\) ⇒ \(a = 8\)? Wait, no, the coefficient of \(x\) in the expanded form from the given box is \(-2a x\) (since inside the parentheses is \(-2x\), multiplied by \(a x\) gives \(-2a x^2\)? Wait, I'm getting confused. Let's start over.

Original expression: \(32x^4 + 24x^3 - 16x^2 - 8x\)

Let's factor out \(x\) first: \(x(32x^3 + 24x^2 - 16x - 8)\)

Now, we need to factor out a number from \(32x^3 + 24x^2 - 16x - 8\) such that:

- \(32x^3 \div \text{number} = \square x^3\)
- \(24x^2 \div \text{number} = \square x^2\)
- \(-16x \div \text{number} = -2x\)
- \(-8 \div \text{number} = -1\)

From the third term: \(-16x \div \text{number} = -2x\) ⇒ \(\text{number} = (-16x) \div (-2x) = 8\). Wait, no: \(-16x \div 8 = -2x\), yes. And \(-8 \div 8 = -1\), which matches. Now check the first term: \(32x^3 \div 8 = 4x^3\)? But in the given parentheses, the first term is \(\square x^3\), second is \(\square x^2\). Wait, no, wait the original expression after factoring out \(x\) is \(x(32x^3 + 24x^2 - 16x - 8)\). Now, if we factor out 8 from the parentheses: \(x \cdot 8(4x^3 + 3x^2 - 2x - 1)\) = \(8x(4x^3 + 3x^2 - 2x - 1)\). But the given form is \([?]x(\square x^3 + \square x^2 - 2x - 1)\). So comparing, \([?] = 8\), \(\square\) (first) = 4, \(\square\) (second) = 3? Wait, no, wait the given parentheses has \(-2x\) and \(-1\), so let's check:

Wait, let's do the factoring properly. Let's factor out the greatest common factor from the entire expression. The GCF of \(32,24,16,8\) is 8, and the GCF of the variables is \(x\) (since the lowest power of \(x\) is \(x^1\)). So GCF is \(8x\). Let's factor out \(8x\):

\(8x(4x^3 + 3x^2 - 2x - 1)\). But the given form is \([?]x(\square x^3 + \square x^2 - 2x - 1)\). So \([?] = 8\), \(\square\) (first) = 4, \(\square\) (second) = 3? Wait, no, wait the original expression is \(32x^4 + 24x^3 - 16x^2 - 8x\). Let's factor out \(8x\):

\(8x(4x^3 + 3x^2 - 2x - 1)\). Yes, because \(8x \cdot 4x^3 = 32x^4\), \(8x \cdot 3x^2 = 24x^3\), \(8x \cdot (-2x) = -16x^2\), \(8x \cdot (-1) = -8x\). Wait, no: \(8x \cdot (-2x) = -16x^2\), yes, and \(8x \cdot (-1) = -8x\), yes. So the first box (the [?]) is 8, the first \(\square\) inside is 4, the second \(\square\) is 3. Wait, but let's check the given form again. The given form is \([?]x(\square x^3 + \square x^2 - 2x - 1)\). So when we factor out \(8x\), it's \(8x(4x^3 + 3x^2 - 2x - 1)\), which matches the form. So the [?] is 8, the first \(\square\) is 4, the second \(\square\) is 3.

Wait, let's verify by expanding: \(8x \cdot 4x^3 = 32x^4\), \(8x \cdot 3x^2 = 24x^3\), \(8x \cdot (-2x) = -16x^2\), \(8x \cdot (-1) = -8x\). Which is exactly the original expression: \(32x^4 + 24x^3 - 16x^2 - 8x\). So that's correct.

Step1: Determine the GCF for the coefficient part after factoring out x

After factoring out \(x\) from each term, we have \(x(32x^3 + 24x^2 - 16x - 8)\). Now, we need to find the GCF of \(32, 24, -16, -8\). The GCF of these numbers is \(8\) (since \(32 \div 8 = 4\), \(24 \div 8 = 3\), \(-16 \div 8 = -2\), \(-8 \div 8 = -1\)). Wait, but in the given parentheses, the third term is \(-2x\) and the fourth term is \(-1\), which matches \(-16x \div 8 = -2x\) and \(-8 \div 8 = -1\). So the GCF of the coefficients \(32,24,16,8\) (after factoring out \(x\)) is \(8\)? Wait, no, \(32,24,16,8\) have GCF 8? Wait, 32 ÷ 8 = 4, 24 ÷ 8 = 3, 16 ÷ 8 = 2, 8 ÷ 8 = 1. But in the parentheses, the third term is \(-2x\), which is \(-16x \div 8 = -2x\), and the fourth term is \(-1\), which is \(-8 \div 8 = -1\). So the first box (the [?]) is 8, the first \(\square\) (inside the parentheses) is 4 (32 ÷ 8), the second \(\square\) is 3 (24 ÷ 8).

Answer:

The first box (?) is \(8\), the first \(\square\) inside the parentheses is \(4\), the second \(\square\) is \(3\). So:

\([8]x(4x^3 + 3x^2 - 2x - 1)\)

So the answers are: \(? = 8\), first \(\square = 4\), second \(\square = 3\).