### Video Transcript

Find the quotient of negative 43𝑥 to the power of seven plus 12𝑥 cubed minus six 𝑥 squared divided by 𝑥.

In order to answer this question, we need to divide each of the terms by 𝑥: negative 43𝑥 to the power of seven divided by 𝑥, 12𝑥 cubed divided by 𝑥, and negative six 𝑥 squared divided by 𝑥.

Using the fact that 𝑥 is the same as 𝑥 to the power of one along with our laws of indices, 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 equals 𝑥 to the power of 𝑎 minus 𝑏 helps us work out that negative 43𝑥 to the power of seven divided by 𝑥 is equal to negative 43𝑥 to the power of six.

In the same way, 12𝑥 cubed divided by 𝑥 is equal to 12𝑥 squared. And finally, negative six 𝑥 squared divided by 𝑥 is equal to negative six 𝑥. Putting these three terms back together gives us an answer of negative 43𝑥 to the power of six plus 12𝑥 squared minus six 𝑥.

This means that the quotient of negative 43𝑥 to the seven plus 12𝑥 cubed minus six 𝑥 squared divided by 𝑥 is negative 43𝑥 to the power of six plus 12𝑥 squared minus six 𝑥.

You will notice that the exponents or indices in our answer are one less than the exponents in the question. This is because we are dividing by 𝑥 or 𝑥 to the power of one. Had we been dividing by 𝑥 squared, we would’ve reduced the exponents by two. And in the same way, dividing by 𝑥 cubed would reduce the exponents by three.