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Wize High School Grade 12 Pre-Calculus Textbook > Polynomial Functions

Applications of Polynomial Functions

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Example: Polynomial Applications

Three consecutive positive integers have a product of 24. What are the numbers?


Let xx, x+1x+1, and x+2x+2 be the 3 consecutive numbers. Then,
x(x+1)(x+2)=24x(x+1)(x+2)=24
Expand & Simplify:
x(x+1)(x+2)=24x3+3x2+2x−24=0\begin{array}{rcl} x(x+1)(x+2)&=&24\\\\ x^3+3x^2+2x-24&=&0 \end{array}

Now we want to factor and solve this equation

Possible Roots: ±1,2,3,4,6,8,12,24\pm1, 2, 3, 4, 6, 8, 12, 24

Test x=2x=2:
x3+3x2+2x−24=(2)3+3(2)2+2(2)−24=8+12+4−24=0\begin{array}{rcl} x^3+3x^2+2x-24&=&(2)^3+3(2)^2+2(2)-24\\\\ &=&8+12+4-24\\\\ &=&0 \end{array}
Therefore, x=2x=2 is a solution.

Let's use synthetic division to divide the polynomial by x=2x=2:
132−242↓2102415120\begin{array}{r|cccc} &1&3&2&-24\\ 2&\downarrow&2&10&24\\\hline &1&5&12&0 \end{array}

The quotient becomes x2+5x+12x^2+5x+12, which has no solutions (use the quadratic formula to see that there's no solutions).

Therefore, the 3 consecutive numbers are 2, 3, 4.
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Example: Polynomial Applications

A box has a height that is twice its width, and a length that is 8m longer than the width. If the volume is 80m380m^3, what are the dimensions of the box?


The volume of a box is V=lwhV=lwh.

Let:
x=width x=\text{width}\ , x>0x>0
2x=height 2x=\text{height}\
x+8=length x+8=\text{length}\
Volume=80\text{Volume}=80

Then,
V=lwh80=x(2x)(x+8)0=2x3+16x2−80\begin{array}{rcl} V&=&lwh\\\\ 80&=&x(2x)(x+8)\\\\ 0&=&2x^3+16x^2-80 \end{array}
Now let's factor then solve this equation

Possible Roots:
±1,2,4,5,8,10,16,20,40,801,2  =  ±12,52,1,2,4,8,10,16,20,40,80\pm\frac{1, 2, 4, 5, 8, 10, 16,20,40,80}{1,2}~~=~~\pm\frac{1}{2}, \frac{5}{2}, 1, 2, 4, 8, 10, 16, 20, 40, 80

Test x=2x=2:
2x3+16x2−80=2(2)3+16(2)2−80=16+64−80=0\begin{array}{rcl} 2x^3+16x^2-80&=&2(2)^3+16(2)^2-80\\\\ &=&16+64-80\\\\ &=&0 \end{array}
So, x=2x=2 is a solution.

Let's use synthetic division to divide the polynomial by x=2x=2:
2160−802↓436802204080\begin{array}{r|cccc} &2&16&0&-80\\ 2&\downarrow&4&36&80\\\hline &2&20&40&80 \end{array}

The quotient becomes 2x2+20x+402x^2+20x+40, which only has negative solutions (use the quadratic formula to see that the only solutions are negative).

Since x>0x>0, then x=2x=2 is our only solution.

Therefore, the dimensions of the box are 2m by 4m by 10m

Practice: Polynomial Applications

A box is constructed by cutting a square of side length 'xx' from each corner and folding up the sides. The length of the box is 3 cm greater than the height and the width of the box is 2 cm greater than the height. If the volume of the box is 12 cubic centimeters, what are the dimensions of the box?

Practice: Polynomial Applications

A 10-foot tall pole is caught in a wind storm. Unfortunately, it can not withstand the wind and breaks into two. The part of the pole that is left standing is the cube-root of the length of the part that broke away. What is the height of the pole left standing?
Extra Practice
Polynomial Applications
Practice: Polynomial Applications
The volume of a fish tank is given by V(x)=x3−16x2+79x−120V(x)=x^3−16x^2+79x−120. If the length of the fish tank is (x−5)(x-5), what are the dimensions for the width and depth?
Polynomial Applications
Practice: Polynomial Applications
The volume of a box is 10x3+65x2+100x+45 cm310x^3+65x^2+100x+45~\text{cm}^3 . The perimeter of the box is 50cm50\text{cm}. What is the surface area?
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Practice: Polynomial Applications
A 10-inch by a 20-inch piece of cardboard is cut and folded into an open lid box. If 'x' inches is cut from each corner, what is an expression for the volume?
Polynomial Applications
Practice: Polynomial Applications
A box is constructed by cutting a square of side length 'xx' from each corner and folding up the sides. The width of the box is 1 cm greater than the height and the height of the box is 2 cm greater than the length. If the volume of the box is 6 cubic centimeters, what are the dimensions of the box?
Remainder Theorem: Applications of Polynomial Functions
The remainder when x3−ax2+bx+3x^3-ax^2+bx+3 is 2 when divided by x−1x-1.
The remainder when 3x3+ax2+bx+13x^3+ax^2+bx+1 is -1 when divided by x+2x+2.
What are the values of aa and bb?
Graphing Polynomial Functions
Write a polynomial function in the form y=a(x−h)n+ky=a(x-h)^n+k, where n∈Nn\in\mathbb{N} and the degree of the polynomial for the following:
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A block has a hollowed out center with the dimensions shown below.
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Practice: Polynomial Applications
The volume of a box is 10x3+65x2+100x+45 cm310x^3+65x^2+100x+45~\text{cm}^3 . What are the dimensions of the box?