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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
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Practice: Polynomial Applications
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Practice: Polynomial Applications
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Practice: Polynomial Applications
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