Question

23y^{10} - 46y^{7} + 68y^{2} + 10y

Explanation:

Step1: Identify the greatest common factor (GCF)

First, we check the coefficients \(23\), \(-46\), \(68\), and \(10\). The GCF of these coefficients: factors of \(23\) are \(1, 23\); factors of \(46\) are \(1, 2, 23, 46\); factors of \(68\) are \(1, 2, 4, 17, 34, 68\); factors of \(10\) are \(1, 2, 5, 10\). The common factor is \(1\) (since \(23\) is prime and doesn't divide \(68\) or \(10\) except for \(1\)). Now for the variable part, the lowest power of \(y\) is \(y^1\) (from \(10y\)). So the GCF is \(y\)? Wait, no, wait: \(23y^{10}\) has \(y^{10}\), \(-46y^7\) has \(y^7\), \(68y^2\) has \(y^2\), \(10y\) has \(y^1\). The lowest exponent of \(y\) is \(1\), but the coefficient GCF: wait, \(23\) and \(46\) have GCF \(23\)? Wait, \(46 = 23\times2\), \(23\) is prime. \(23\) and \(68\): \(68\div23\) is not integer. \(23\) and \(10\): no. So coefficient GCF is \(1\). Wait, but maybe I made a mistake. Wait, \(23y^{10}-46y^7 + 68y^2+10y\). Let's check coefficients again: \(23\), \(-46\) (which is \(23\times(-2)\)), \(68\), \(10\). So \(23\) is a factor of first two terms, but not the last two. So actually, the GCF of all terms is \(1\), so we can't factor out a numerical coefficient or a higher power of \(y\) than \(y^1\), but \(y\) is a factor? Wait, \(23y^{10}=y\times23y^9\), \(-46y^7 = y\times(-46y^6)\), \(68y^2 = y\times68y\), \(10y = y\times10\). So we can factor out \(y\):

Step2: Factor out \(y\)

\(23y^{10}-46y^7 + 68y^2+10y = y(23y^9 - 46y^6 + 68y + 10)\)

Wait, but let's check the coefficients inside the parentheses: \(23y^9 - 46y^6 + 68y + 10\). Is there a common factor here? \(23\) and \(46\) have GCF \(23\), but \(68\) and \(10\) don't. So we can't factor further with integer coefficients. So the factored form is \(y(23y^9 - 46y^6 + 68y + 10)\) or we can also note that the original polynomial can be factored by taking \(y\) as the GCF.

Wait, maybe I messed up the coefficient GCF. Let's re-express:

Original polynomial: \(23y^{10}-46y^7 + 68y^2+10y\)

Let's list the terms:

1. \(23y^{10}\)
2. \(-46y^7 = -2\times23y^7\)
3. \(68y^2\)
4. \(10y\)

So term 1 and 2 have a common factor of \(23y^7\), but terms 3 and 4 don't share that. So the overall GCF for all four terms is \(y\) (since each term has at least \(y^1\)) and the coefficient GCF is \(1\) (since \(23\), \(-46\) (which is \(23\times -2\)), \(68\), \(10\) have no common factor other than \(1\)). So factoring out \(y\):

\(y(23y^9 - 46y^6 + 68y + 10)\)

Alternatively, maybe the problem is to factor the polynomial, so the factored form is \(y(23y^9 - 46y^6 + 68y + 10)\) or we can check if there's a mistake in the problem, but assuming we need to factor, that's the way.

Answer:

\(y(23y^9 - 46y^6 + 68y + 10)\)