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### Course: Algebra (all content) > Unit 10

Lesson 9: Polynomials word problems# Polynomial word problem: rectangle and circle area

Sal writes a binomial to express the difference between the area of a rectangle and the area of a circle. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- why do they call it a triominal?(32 votes)
- Polynomials" refer to mathematical expressions that contain multiple terms (poly = multiple, nomials = terms/numbers (roughly)). "Monomial" refers to a polynomial that has only a single (mono) term.(32 votes)

- is there something bigger than trinominal?(7 votes)
- Well, a polynomial can have more than three terms, but there are no more specific terms - after three terms, it's just a polynomial.(11 votes)

- What is the equation of the area of a cirle(6 votes)
- pi * (r^2), where "r" is the radius of the circle and "pi" is, well, 3.14159265358979323846...(12 votes)

- how doe polynomials help in actual real life?(6 votes)
- Polynomials are like an express way of calculating financial planning, construction of materials, expense budgeting, and helps greatly in science. (finding initial velocities, gravitational acceleration, etc.)

http://www.ehow.com/info_10070112_polynomials-used-daily-life.html(7 votes)

- What is the largest number of real roots that a fourth degree polynomial could have? What is the smallest number?(4 votes)
- it could potentially have 4 real roots, in some cases it can have none. this is because there must be four solutions, real or imaginary. they may be all real or all imaginary as the exponent is even so it comes from a high positive to a high positive (imagine the graph) hope this helps(2 votes)

- At4:08, Sal's final expression (2rp - 4πr^2) had three terms : r, p, and r^2. But didn't the question ask for a binomial, which has two terms? I'm confused....(2 votes)
- This expression has two terms. I think you counted the types of variables in the expression instead of the terms.

A term is an expression that doesn't have the operations of addition and subtraction.

2rp<---One term

4πr^2<---Second term

There are two terms in this expression; therefore, it's a binomial. I hope this helped!(3 votes)

- Ok I'm 17 in the 11th grade and new to khan and I'm seriously bad at math...like SERIOUSLY BAD! when I look at math equations all I see is a bunch of letters and numbers. I have a really hard time remembering what I learn too. :/ it's like I learn it then when I have to do it again the next day I have to re-learn everything. I'm pretty visual. Is there like a special way to maybe get a video chat thing with one of khan academy's teachers so they could like tutor me kinda if I need help?(2 votes)
- How do you find the degree of a polynomial(3 votes)
- The degree of a polynomial expression is equal to the value of the highest exponent. I.e. x^2 + 3x^4 +4 the degree of this polynomial is 4 because of the term 3x^4(2 votes)

- Would it be incorrect to instead write it as |2rp-4πr^2| or |4πr^2-2rp| once you know the equation for each area?(2 votes)
- Why did Sal put the area of the rectangle and the 14r and the area of the circle in an inequality?(2 votes)
- It is because the problem said that p>7r, it did not say specifically what it was equivalent to. Hope this helps!(1 vote)

## Video transcript

Write a binomial to express
the difference between the area of a rectangle with
length p and width 2r and the area of a
circle with diameter 4r. And they tell us that
p is greater than 7r. So let's first
think about the area of a rectangle with
length p and width 2r. So this is our
rectangle right here. It has a length of p and
it has a width of 2r. So what's its area? Well, it's just going to be
the length times the width. So the area here is going to be
p-- or maybe I should say 2rp. This is the length
times the width, or the width times the length. So area is equal to
2rp for the rectangle. Now, we also want to find the
difference between this area and the area of a
circle with diameter 4r. So what's the area of
the circle going to be? So let me draw our
circle over here. So our circle looks like that. Its diameter is 4r. How do we figure out
the area of a circle? Well, area is equal
to pi r squared for a circle, where
r is a radius. They gave us the diameter. The radius is half of that. So the radius here is going to
be half this distance, or 2r. So the area of our circle
is going to be pi times 2r, the whole thing squared. This is the radius, right? So we're squaring
the entire radius. So this is going to be equal
to pi times 4 times r squared. I'm just squaring
each of these terms. Or if we were to change the
order, the area of the circle is equal to 4 pi r squared. And we want to find
the difference. So to find a difference,
It's helpful-- just so we don't end up
with a negative number-- to figure out which of
these two is larger. So they're telling us that
p is greater than 7 r. So let's think about this. If p is greater than 7r, then
2-- let me write it this way. We know that p is
greater than 7r. So if we're going to multiply
both sides of this equation by 2rr-- and 2r
is positive, we're dealing with positive
distances, positive lengths-- so if we multiply both sides
of this equation by 2r, it shouldn't change
the equation. So multiply that by 2r, and
then multiply this by 2r. And then our equation
becomes 2rp is greater than 14r squared. Now, why is this interesting? Actually, why did I even
multiply this by 2r? Well, that's so that
this becomes the same as the area of the rectangle. So this is the area
of the rectangle. And what's 14r squared? Well, 4 times pi, is going to
get us something less than 14. This is less than 14. So this is 4 pi is less than 14. 14 is 4 times 3 and 2--
let me put it this way. 4 times 3.5 is equal to 14. Right? So 4 times pi, which
is less than 3.5, is going to be less than 14. So we know that
this over here is larger than this
quantity over here. It's larger than 4 pi r squared. And so we know
that this rectangle has a larger area
than the circle. So we can just subtract
the circle's area from the rectangle's area
to find the difference. So the difference is going to
be the area of the rectangle, which we already
figured out is 2rp. And we're going to subtract from
that the area of the circle. The area of the circle
is 4 pi r squared. So hopefully that made sense. And one point I want to clarify. I gave the equation of the area
of a circle to be pi r squared. And then we said that the radius
is actually 2r in this case. So I substituted 2r for r. Hopefully that
doesn't confuse you. This r is the general
term for any radius. They later told us
that the actual radius is 2 times some letter r. So I substitute that
into the formula. Anyway, hopefully you
found that useful.