Question 15, 6.1.51 Factor the greatest common factor (GCF) from the following polynomial. \( 5 x^{4} y^{3}-10 x^{5} y^{4} \)

To factor the greatest common factor (GCF) from the polynomial \( 5 x^{4} y^{3} - 10 x^{5} y^{4} \), we first identify the GCF of the coefficients and the variables. The GCF of the coefficients \( 5 \) and \( -10 \) is \( 5 \). For the variables, the lowest power of \( x \) is \( x^{4} \), and for \( y \), it is \( y^{3} \). Thus, the GCF is \( 5 x^{4} y^{3} \). Now, we can factor it out from the polynomial: \[ 5 x^{4} y^{3} (1 - 2xy) \] So, the final factored form of the polynomial is \( 5 x^{4} y^{3} (1 - 2xy) \). Knowing how to identify and factor the GCF is essential in mathematics as it lays the groundwork for simplifying expressions and solving equations more easily. It’s like finding the biggest common friend among a group—they make everything easier to manage! When factoring polynomials, one common mistake is forgetting to include all variable factors in the GCF. Always double-check that you've included the lowest powers of each variable present in the terms. It’s a typical slip that can lead to errors down the line!