1. Consider the function f(x) = x3 + 7x2 + 2x - 40 .
a. Find the possible rational roots of f(x) = 0.
b. Use synthetic division to divide f(x) by x - 2. Show all your work. Use your answer to explain whether or not x – 2 is a factor of f(x).
c. Factor f(x) completely. Show all your work.
d. Sketch the graph of f(x) by plotting the x-intercepts, finding and plotting additional points on the graph, and using what you know about the graph’s end behavior.
Why don't you start with b)
You would see that the division is exact , thus x-2 is a factor
Your synthetic division should have a result of x^2 + 9x + 20
so it fully factors to (x-2)(x+5)(x+4)
Once factored, the roots are simple to see.
btw, the order of the questions is poorly presented,
question a) and c) should be interchanged.
a. To find the possible rational roots of f(x) = 0, we can use the Rational Root Theorem. According to the theorem, any possible rational root of a polynomial is of the form p/q, where p is a factor of the constant term (in this case, -40) and q is a factor of the leading coefficient (in this case, 1).
The factors of 40 are: ±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40.
The factors of 1 are: ±1.
Therefore, the possible rational roots of f(x) = 0 are:
±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40.
b. To use synthetic division to divide f(x) by x - 2, we set up the synthetic division table:
<pre>
| 1 7 2 -40
-----------------------------
2 |
|________
</pre>
Now, we bring down the first coefficient, which is 1:
<pre>
| 1 7 2 -40
-----------------------------
2 | 1
|________
</pre>
Next, we multiply the divisor (2) by the value in the bottom row and write it under the next coefficient (7):
<pre>
| 1 7 2 -40
-----------------------------
2 | 1 2
|________
</pre>
Next, we add the numbers in the second column:
<pre>
| 1 7 2 -40
-----------------------------
2 | 1 2
|__________
1 9
</pre>
Next, we multiply the divisor (2) by the value in the bottom row and write it under the next coefficient (2):
<pre>
| 1 7 2 -40
-----------------------------
2 | 1 2
|__________
1 9
-----------
1 11
</pre>
Lastly, we multiply the divisor (2) by the value in the bottom row and write it under the next coefficient (-40):
<pre>
| 1 7 2 -40
-----------------------------
2 | 1 2
|__________
1 9
-----------
1 11
22
-----------
1 -18
</pre>
The numbers in the bottom row represent the coefficients of the quotient of the division. So, the quotient is 1x^2 + 11x - 18. Hence, x - 2 is not a factor of f(x), as the quotient is not zero.
c. To factor f(x) completely, we can use the Rational Root Theorem to find the roots and then use synthetic division to find the remaining factor(s).
Since the leading coefficient is 1, we can apply the Rational Root Theorem to find the roots of f(x). From part (a), the possible rational roots are: ±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40.
By trying different possible rational roots, we find that x = -2 is a root of f(x). Therefore, (x + 2) is a factor.
Using synthetic division:
<pre>
| 1 7 2 -40
-----------------------------
-2 | 1 7
|_________
1 -5
-----------
1 2
0
</pre>
The remaining factor is 1x + 2 = x + 2.
Therefore, f(x) = (x + 2)(x^2 - 5).
To factor x^2 - 5 further, we can use difference of squares:
f(x) = (x + 2)(x - √5)(x + √5).
d. To sketch the graph of f(x), we need to find the x-intercepts, additional points, and understand the end behavior.
To find the x-intercepts, we set f(x) = 0:
(x + 2)(x - √5)(x + √5) = 0
So the x-intercepts are x = -2, x = √5, and x = -√5.
To find additional points on the graph, we can select some arbitrary values of x and calculate the corresponding y-values. For example, when x = 0, we have:
f(0) = (0 + 2)(0 - √5)(0 + √5) = 2(0 - 2)(0 + 2) = -16
So we have the point (0, -16) on the graph.
To understand the end behavior of the graph, we look at the leading term of f(x):
f(x) = x^3 + 7x^2 + 2x - 40
As x approaches negative infinity, the leading term x^3 dominates, and the graph will approach negative infinity.
As x approaches positive infinity, the leading term x^3 still dominates, and the graph will approach positive infinity.
Using all this information, we can sketch the graph of f(x) by plotting the x-intercepts (-2, √5, -√5), the additional point (0, -16), and understanding the end behavior.
a. Finding the possible rational roots of f(x) = 0 involves applying the Rational Root Theorem. According to the theorem, any rational root of the function can be expressed as a fraction in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is -40 and the leading coefficient is 1.
To find the possible rational roots, we need to find the factors of 40 and list all the possible combinations of p/q, where p is a factor of 40 and q is a factor of 1.
The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, and 40.
The factors of 1 are: 1.
Now we can list the possible combinations of p/q: ±1/1, ±2/1, ±4/1, ±5/1, ±8/1, ±10/1, ±20/1, ±40/1.
Therefore, the possible rational roots of f(x) = 0 are: ±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40.
b. To divide f(x) = x^3 + 7x^2 + 2x - 40 by x - 2 using synthetic division, we set up the division as follows:
2 | 1 7 2 -40
__________________
Multiply 2 by 1: 2
Add result to 7: 9
Multiply 2 by 9: 18
Add result to 2: 20
Multiply 2 by 20: 40
Add result to -40: 0
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The final result is 1x^2 + 9x + 20 with a remainder of 0. Therefore, x - 2 is a factor of f(x) since the remainder is 0.
c. To factor f(x) completely, first, determine the roots of the equation f(x) = 0. From part b, we know that x - 2 is a factor of f(x). Therefore, we can write f(x) as (x - 2)(ax^2 + bx + c), where ax^2 + bx + c is a quadratic expression.
Now, divide f(x) by (x - 2) using long division or synthetic division to find the quadratic expression:
2 | 1 7 2
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Multiply 2 by 1: 2
Add result to 7: 9
Multiply 2 by 9: 18
Add result to 2: 20
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The quadratic expression is 1x^2 + 9x + 20. We can further factor this quadratic expression as (x + 4)(x + 5).
Therefore, f(x) = (x - 2)(x + 4)(x + 5).
d. To sketch the graph of f(x), follow these steps:
1. Plot the x-intercepts by setting f(x) = 0 and solving for x:
- (x - 2) = 0 => x = 2 (x-intercept at x = 2)
- (x + 4) = 0 => x = -4 (x-intercept at x = -4)
- (x + 5) = 0 => x = -5 (x-intercept at x = -5)
2. Find additional points on the graph by substituting values for x and evaluating f(x). For example, you can choose x-values like x = -3, -2, -1, 0, 1, 3 and calculate the corresponding y-values using f(x).
3. Plot the points obtained from step 1 and step 2 on a graph.
4. Determine the end behavior of the graph by observing the leading term of the polynomial function. In this case, the leading term is x^3.
- As x approaches negative infinity, f(x) approaches negative infinity because the leading term dominates.
- As x approaches positive infinity, f(x) approaches positive infinity because the leading term dominates.
Using the x-intercepts, additional points, and the knowledge about end behavior, sketch the graph of f(x).