Simplify: StartFraction x cubed plus 5 x squared minus 20 over x plus 3 EndFraction
Move values to the blanks to complete the equation.
Response area with 4 blank spaces
Blank space 1 empty
Blank space 5 empty
Blank space 7 empty
Blank space 8 empty
Answer options with 7 options.
-6
-2
1
2
8
24
52
The simplified expression is:
Blank space 1: x
Blank space 5: -3
Blank space 7: x
Blank space 8: 5
Answer: (x - 3)(x^2 + 3x + 5)
To simplify the expression StartFraction x cubed plus 5 x squared minus 20 over x plus 3 EndFraction, we can factor the numerator.
First, let's factor out an "x" from each term in the numerator:
x(x^2 + 5x - 20)
Next, let's factor the quadratic expression inside the parentheses:
x(x - 4)(x + 5)
Now, the simplified expression is:
StartFraction x(x - 4)(x + 5) over x + 3 EndFraction
Thus, the values to complete the equation are as follows:
Blank space 1: x(x - 4)(x + 5)
Blank space 5: /
Blank space 7: x + 3
Blank space 8: empty
Answer options: none of the above
To simplify the given expression, we need to divide each term by the denominator (x + 3). Let's divide each term individually:
Term 1: x^3 divided by (x + 3)
When dividing x^3 by (x + 3), we get x^2 (as the exponent is decreased by 1).
Term 2: 5x^2 divided by (x + 3)
When dividing 5x^2 by (x + 3), we get 5x (as the exponent is decreased by 1).
Term 3: -20 divided by (x + 3)
When dividing -20 by (x + 3), we get -20/(x + 3).
Now we can rewrite the expression:
(x^3 + 5x^2 - 20) / (x + 3) = x^2 + 5x - 20/(x + 3)
So, the simplified expression is x^2 + 5x - 20/(x + 3).
Let's substitute the answer options into the equation to find the correct one:
Option 1: Substitute -6 into the equation:
(-6)^2 + 5(-6) - 20/(-6 + 3) = 36 - 30 - 20/(-3) = 6 - 20/(-3) = 6 + 20/3 = 18/3 + 20/3 = 38/3 ≠ 0
Option 2: Substitute -2 into the equation:
(-2)^2 + 5(-2) - 20/(-2 + 3) = 4 - 10 - 20/1 = -6 - 20 = -26 ≠ 0
Option 3: Substitute 1 into the equation:
(1)^2 + 5(1) - 20/(1 + 3) = 1 + 5 - 20/4 = 6 - 5 = 1 ≠ 0
Option 4: Substitute 2 into the equation:
(2)^2 + 5(2) - 20/(2 + 3) = 4 + 10 - 20/5 = 14 - 4 = 10 ≠ 0
Option 5: Substitute 8 into the equation:
(8)^2 + 5(8) - 20/(8 + 3) = 64 + 40 - 20/11 = 104 - 20/11 ≠ 0
Option 6: Substitute 24 into the equation:
(24)^2 + 5(24) - 20/(24 + 3) = 576 + 120 - 20/27 = 696 - 20/27 ≠ 0
Option 7: Substitute 52 into the equation:
(52)^2 + 5(52) - 20/(52 + 3) = 2704 + 260 - 20/55 = 2964 - 20/55 ≠ 0
None of the answer options simplifies the expression to 0. Therefore, the correct option is not among the given answer options.