Question

which of the following is the equivalent form of $8x^{10}y^7 - 12x^{11}y^5 + 32x^8y^3$ $2x^8y^3(4x^2y^4 - 6x^3y^2 + 16)$ $4x^8y^3(2x^2y^4 - 3x^3y^2 + 8)$ $4x^8y^3(2x^2y^4 - 3x^3y^2 + 8)$ $8x^8y^3(x^2y^4 - 3x^3y^2 + 4)$

Explanation:

Step1: Find the GCF of coefficients and variables

First, find the greatest common factor (GCF) of the coefficients \(8\), \(-12\), and \(32\). The GCF of \(8\), \(12\), and \(32\) is \(4\). For the \(x\)-terms: \(x^{10}\), \(x^{11}\), \(x^{8}\), the lowest power is \(x^{8}\). For the \(y\)-terms: \(y^{7}\), \(y^{11}\), \(y^{3}\), the lowest power is \(y^{3}\). So the GCF of the polynomial is \(4x^{8}y^{3}\).

Step2: Divide each term by the GCF

Divide \(8x^{10}y^{7}\) by \(4x^{8}y^{3}\): \(\frac{8x^{10}y^{7}}{4x^{8}y^{3}} = 2x^{2}y^{4}\)

Divide \(-12x^{11}y^{11}\) by \(4x^{8}y^{3}\): \(\frac{-12x^{11}y^{11}}{4x^{8}y^{3}}=- 3x^{3}y^{8}\)? Wait, no, wait, original term is \(-12x^{11}y^{11}\)? Wait, no, the original polynomial is \(8x^{10}y^{7}-12x^{11}y^{11}+32x^{8}y^{3}\)? Wait, no, looking back, the options have \(2x^{2}y^{4}\), \(4x^{8}y^{3}\), etc. Wait, maybe I made a mistake in the exponents. Let's re - check the original polynomial: \(8x^{10}y^{7}-12x^{11}y^{11}+32x^{8}y^{3}\)? Wait, no, the third term is \(32x^{8}y^{3}\), the second term: maybe it's \( - 12x^{11}y^{11}\) or maybe a typo? Wait, looking at the options, let's check the third option: \(4x^{8}y^{3}(2x^{2}y^{4}-3x^{3}y^{2}+8)\). Let's multiply this out:

\(4x^{8}y^{3}\times2x^{2}y^{4}=8x^{10}y^{7}\)

\(4x^{8}y^{3}\times(-3x^{3}y^{2})=-12x^{11}y^{5}\)? Wait, no, the original second term is \(-12x^{11}y^{11}\)? Wait, maybe the original polynomial's second term is \(-12x^{11}y^{5}\)? Wait, no, the options: let's check the third option: \(4x^{8}y^{3}(2x^{2}y^{4}-3x^{3}y^{2}+8)\)

Multiply \(4x^{8}y^{3}\times2x^{2}y^{4}=8x^{10}y^{7}\)

Multiply \(4x^{8}y^{3}\times(-3x^{3}y^{2})=-12x^{11}y^{5}\)? No, that's not matching. Wait, maybe the original polynomial is \(8x^{10}y^{7}-12x^{11}y^{5}+32x^{8}y^{3}\). Then:

GCF of \(8\), \(12\), \(32\) is \(4\). GCF of \(x\): \(x^{8}\), GCF of \(y\): \(y^{3}\). So GCF is \(4x^{8}y^{3}\)

Divide \(8x^{10}y^{7}\) by \(4x^{8}y^{3}\): \(2x^{2}y^{4}\)

Divide \(-12x^{11}y^{5}\) by \(4x^{8}y^{3}\): \(-3x^{3}y^{2}\)

Divide \(32x^{8}y^{3}\) by \(4x^{8}y^{3}\): \(8\)

So the factored form is \(4x^{8}y^{3}(2x^{2}y^{4}-3x^{3}y^{2}+8)\), which is the third option.

Wait, let's re - do the factoring properly. Let's take the polynomial \(8x^{10}y^{7}-12x^{11}y^{5}+32x^{8}y^{3}\) (assuming a typo in the exponent of \(y\) in the second term).

1. Find GCF of coefficients: \(\gcd(8,12,32)=4\)
2. Find GCF of \(x\) - terms: \(x^{8}\) (since \(8\leq10\leq11\))
3. Find GCF of \(y\) - terms: \(y^{3}\) (since \(3\leq5\leq7\))

So GCF is \(4x^{8}y^{3}\)

Now, divide each term by \(4x^{8}y^{3}\):

• For \(8x^{10}y^{7}\): \(\frac{8x^{10}y^{7}}{4x^{8}y^{3}} = 2x^{2}y^{4}\)
• For \(-12x^{11}y^{5}\): \(\frac{-12x^{11}y^{5}}{4x^{8}y^{3}}=-3x^{3}y^{2}\)
• For \(32x^{8}y^{3}\): \(\frac{32x^{8}y^{3}}{4x^{8}y^{3}} = 8\)

So the factored form is \(4x^{8}y^{3}(2x^{2}y^{4}-3x^{3}y^{2}+8)\), which is the third option.

Answer:

\(4x^{8}y^{3}(2x^{2}y^{4}-3x^{3}y^{2}+8)\) (the third option, i.e., the option with \(4x^{8}y^{3}(2x^{2}y^{4}-3x^{3}y^{2}+8)\))