Answer:
The polynomial is [tex]x^3+2x-3[/tex].
Step-by-step explanation:
The quotient stands for division, so when we are told that the quotient of [tex]x^5-3x^3-3x^2-10x+15[/tex] and a polynomial, we have to think of a division which result is [tex]x^2-5[/tex]. The goal of this exercise is to find the value of such polynomial that is divisor.
Solving for the divisor
We can write the quotient in a general way using proper naming as follows:
[tex]\cfrac{\text{dividend}}{\text{divisor}}=\text{result}[/tex]
Solving for the divisor we have
[tex]\text{dividend}=\text{divisor} \times\text{result}\\\cfrac{\text{dividend}}{\text{result}}=\text{divisor}[/tex]
So the goal is to work with that division to find the polynomial, that is
[tex]\cfrac{x^5-3x^3-3x^2-10x+15}{x^2-5}[/tex]
Long division
In order to set up for long division, we have to realize that the dividend or numerator has not the [tex]x^4[/tex] term, so we can fill it with [tex]0x^4[/tex]
So we will have
[tex]x^5+0x^4-3x^3-3x^2-10x+15[/tex]
Please check the attached image file to see the steps for the long division that will be explained here in text.
The first step is to divide always the first term of the dividend that is the [tex]x^5[/tex] by the first term of the polynomial we are dividing by, that is [tex]x^2[/tex], so we get
[tex]\cfrac{x^5}{x^2} =x^3[/tex]
We write that on top and then we multiply it times the polynomial we are dividing by that is
[tex]x^3(x^2-5) = x^5-5x^3[/tex]
We write that below the dividend and proceed subtracting each column, that will give us
[tex]2x^3-3x^2-10x+15[/tex]
So we can continue with the division, just looking at first terms we get
[tex]\cfrac{2x^3}{x^2}=2x[/tex]
Then we continue multiplying
[tex]2x(x^2-5) = 2x^3-10x[/tex]
And we write that below as can be seen on the attached image, so the subtraction of each column give us
[tex]-3x^2+15[/tex]
Finally we can work with the last division of first terms.
[tex]\cfrac{-3x^2}{x^2}=-3[/tex]
And we write that -3 on top, and we multiply
[tex]-3(x^2-5) = -3x^2+15[/tex]
And work with the last division and we get 0. Getting 0 is a way to verify that there is no remainder and that our work is correct for the quotient.
Lastly the expression on top is the original polynomial of the quotient, we can conclude that the polynomial is [tex]x^3+2x-3[/tex]
