Standardized Test Practice
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Recall the formula of a rectangular prism.
How many zeros does a nth degree polynomial have?
What are the possible rational roots of the given polynomial?
h^3+2h^2-3h-864=0
Number of solutions: 3
Explanation: See solution.
8* 12* 9
Let h be the height of the given prism. We will rewrite length l and width w using h.
Let's recall the Fundamental Theorem of Algebra.
Recall that any rational root for h(x)=0 has the form ± p q, where p is a factor of the constant term and q is a factor of the leading coefficient. Since q is equal to 1 we need to consider only factors of p. Let's substitute the positive factors of -864 into the polynomial until we find the first zero.
| Positive factors of -864 | h^3+2h^2-3h-864? =0 | Is it a zero? |
|---|---|---|
| 1 | 1^3+2( 1)^2-3( 1)-864? =0 | -864≠0 * |
| 2 | 2^3+2( 2)^2-3( 2)-864? =0 | -854≠0 * |
| ... | ... | ... |
| 8 | 8^3+2( 8)^2-3( 8)-864? =0 | -248≠0 * |
| 9 | 9^3+2( 9)^2-3( 9)-864? =0 | 0= 0 ✓ |
We see above that a zero occurs at h=9. Let's use synthetic division to depress the polynomial and see if there are some other possible solutions.
Bring down the first coefficient
Multiply the coefficient by the divisor
Add down
Multiply the coefficient by the divisor
Add down
Multiply the coefficient by the divisor
Add down
The above are the coefficients of the depressed polynomial. h^3+2h^2-3h-864=0 ⇕ (h-9) ( h^2+11h+96 )=0 Now the depressed is a quadratic polynomial, where a= 1, b= 11, and c= 96. This means we can use the Quadratic Formula to find its zeros.
Substitute values
Calculate power
a * 1=a
(- a)b = - ab
Subtract term
The discriminant of this quadratic expression is negative, so we do not have any more radical solutions to our equation. Now, since we know that the height equals 9, we can find the length and width. l=9-1=8 w=9+3=12 Therefore, the given prism has dimensions of 8* 12* 9.