Video Transcript
What is the width of a rectangle
whose area is negative π₯ to the fourth power plus 20π₯ squared plus 16π₯ square
centimeters and whose length is negative π₯ squared plus four π₯ plus four
centimeters?
Letβs begin by sketching a diagram
of a rectangle with the given length and area. We recall that the area of a
rectangle is given by the product of its length and width. This means we can determine the
width of this rectangle by dividing its area by its length. In this case, we will be dividing a
fourth-degree polynomial by a quadratic polynomial. We can do this by using polynomial
long division. When setting up our long division,
we notice that the fourth-degree polynomial does not have a cubic term or a
constant. In this situation, weβll insert a
zero for each missing term. This allows us to have a column for
each degree term in standard form order for our future computations.
To begin, we divide the leading
term of the dividend by the leading term of the divisor to get π₯ squared. We add this to the quotient and
then subtract π₯ squared times the divisor. We find the product of π₯ squared
in the divisor is negative π₯ to the fourth power plus four π₯ cubed plus four π₯
squared. Then, we subtract this expression
from the dividend. We must be careful to subtract each
term, not just the first term. It may be helpful to first
distribute the negative through the parentheses then combine like terms. This looks like adding π₯ to the
fourth power, subtracting four π₯ cubed, and subtracting four π₯ squared. This gives us the new dividend,
negative four π₯ cubed plus 16π₯ squared plus 16π₯.
As long as the degree of the
dividend is greater than or equal to the degree of the divisor, we apply this
process again. So, we start again with the leading
term of the new dividend divided by the leading term of the divisor. The quotient of negative four π₯
cubed divided by negative π₯ squared is four π₯. So, we add four π₯ to the
quotient. Then, we multiply four π₯ with the
divisor. The result is negative four π₯
cubed plus 16π₯ squared plus 16π₯, which we need to subtract from the new
dividend. At this step, we can distribute the
negative sign and then combine like terms. The result is a remainder of
zero. So, the quotient is π₯ squared plus
four π₯.
Having taken the quotient of the
area divided by the length, we found the width of the rectangle to be π₯ squared
plus four π₯ centimeters.
