a What is the relationship between the degree of the polynomial and its linear factors?
B
b If the length is the greatest dimension, then it will be the greatest linear factor.
C
c Volume is the depth multiplied by the length and by the width.
A
aHow many linear factors? Three
What are they? (x-1), (x+3), and (2x+5)
B
bWhich linear factor represents the 13 ft length? (2x+5)
Tank Dimensions: 13 ft by 3 ft by 7 ft
C
cVolume of the tank: 273 ft^3
Does the x value substituted in the given polynomial give the same volume? Yes.
Practice makes perfect
a We are looking for a total of three linear factors. This is because the polynomial is degree 3 and there are three dimensions (depth, length, and width) for the tank. We can start by dividing the given polynomial using synthetic division.
2x^3+9x^2+4x-15
Since there are no missing terms, we do not need to rewrite the polynomial. Remember that the general form of the synthetic division divisor must be x-a. Since the linear factor for depth (x-1) is already written in this form, we are ready to go.
rl IR-0.15cm r 1 & |rr 2 &9 &4 &-15 & & & & c & & &
Bring down the first coefficient
rl IR-0.15cm r 1 & |rr 2 &9 &4 &-15 & & & & c 2 & & &
Multiply the coefficient by the divisor
rl IR-0.15cm r 1 & |rr 2 &9 &4 &-15 &2 & & & c 2 & & &
Add down
rl IR-0.15cm r 1 & |rr 2 &9 &4 &-15 &2 & & & c 2 & 11 & &
â–Ľ
Multiply the coefficient by the divisor
rl IR-0.15cm r 1 & |rr 2 &9 &4 &-15 &2 &11 & & c 2 & 11 & &
Add down
rl IR-0.15cm r 1 & |rr 2 & 9 &4 &-15 &2&11& & c 2 & 11 &15 &
Multiply the coefficient by the divisor
rl IR-0.15cm r 1 & |rr 2 & 9 &4 &-15 &2&11&15 & c 2 & 11 & 15 &
Add down
rl IR-0.15cm r 1 & |rr 2 & 9 &4 &-15 &2&11&15 & c 2 & 11 & 15 &0
The quotient is a polynomial of degree 2, with the above coefficients. The remainder is 0.
Quotient & Remainder
2x^2+11x+15 & 0
Here we have a quadratic trinomial of the form ax^2+bx+c, where |a| ≠1 and there are no common factors. To factor this expression, we will rewrite the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b.
2x^2+11x+15
We have that a= 2, b=11, and c=15. There are now three steps we need to follow in order to rewrite the above expression.
Find a c. Since we have that a= 2 and c=15, the value of a c is 2* 15=30.
Find factors of a c. Since a c=30, which is positive, we need factors of a c to have the same sign — both positive or both negative — in order for the product to be positive. Since b=11, which is positive, those factors will need to be positive so that their sum is positive.
We get two additional linear factors (2x+5) and (x-3). Along with the one given (x-1), that makes three linear factors in total.
b Length is the greatest dimension of the tank. Therefore, for any x value, this linear factor will be the largest. This means that the (2x+5) factor represents length because it doubles the value and adds 5 while the other two only subtract from it. Since we know the length is 13 ft, we can set the two equal to each other and solve for x.
