Video Transcript
Simplify 𝑥 to the fourth power
plus seven 𝑥 cubed minus nine 𝑥 squared all over 𝑥 squared.
In this question, we are asked to
simplify the division of a polynomial by a monomial. And we can do this by dividing each
term in the numerator by the denominator separately. It is worth reiterating that we do
need to be careful to include any signs in the terms. We obtain 𝑥 to the fourth power
over 𝑥 squared plus seven 𝑥 cubed over 𝑥 squared minus nine 𝑥 squared over 𝑥
squared.
We can now simplify each term by
using the quotient rule, which tells us that 𝑥 to the power of 𝑎 over 𝑥 to the
power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. It is worth reiterating that this
does not hold true if the value of 𝑥 is equal to zero. We can apply this to each term by
taking the coefficient out and then subtracting the exponents of 𝑥. We get 𝑥 to the power of four
minus two plus seven 𝑥 to the power of three minus two minus nine 𝑥 to the power
of two minus two.
We can evaluate each of the
exponents to get the expression 𝑥 squared plus seven 𝑥 to the first power minus
nine 𝑥 to the zeroth power. We can simplify this further by
recalling two of our exponent laws. We know that raising any value to
the first power leaves it unchanged and that raising any nonzero number to the
zeroth power gives us one. Since we assumed 𝑥 was nonzero
when we used the quotient rule, we can use this to rewrite the expression as 𝑥
squared plus seven 𝑥 minus nine times one, which is the same as subtracting
nine.
