Answer:
a. A = x² - 36x + 288
b. If x = 2; A = 220 in²
Step-by-step explanation:
<h2>a.</h2>
In this scenario, 'x' represents the width of the border. A "uniform border" means that the border has the same width all around.
The area of the paper that can be painted does not include the border. Since the border is on the original rice paper (not added to it), you subtract 'x' from each side of the rice paper.
The area of a rectangle is A = lw (Area is length times width). <u>Each length and width is reduced by 'x'.</u>
If the original rice paper's area is: 24 X 12, then,
the Area that can be painted is:
A = (24 - x) (12 - x)
A polynomial in standard form means the most expanded version of an expression, ordering the terms (numbers) from greatest to least degree. (The degree of a term is the value of its exponent).
Expand the expression. Multiply the numbers from each bracket in the order FOIL (first, outside, inside last):
A = (24 - x) (12 - x)
= 288 - 24x - 12x + x² Collect like terms (-24x - 12x = -36x)
= 288 - 36x + x² Rearrange ordering greatest to least exponent value
= x² - 36x + 288 Standard form polynomial
<h2>b.</h2>
The width of the margin is the same thing as the width of the border. Since 'x' is the width of the border,
x = 2
Using either the expanded or factored equation for area of that can be painted, substitute 'x' for 2.
I will use the factored form.
A = (24 - x) (12 - x) Substitute x for 2
= (24 - 2) (12 - 2) Solve inside each bracket first by subtracting
= (22) (10) Multiply
= 220 Area that can be painted
Remember to include the units: 220 in²
Therefore if the margin is 2 inches wide, then the area that can be painted in 220 square inches.
, and then we are going to factor backwards.
is a factor of our polynomial.
, so lets set 
is a factor of our polynomial.