Write the polynomial f(x)=x^4-10x^3+25x^2-40x+84. In factored form
1 answer:
<h2>Steps:</h2>
So firstly, to factor this we need to first find the potential roots of this polynomial. To find it, the equation is , with p = the factors of the constant and q = the factors of the leading coefficient. In this case:
Next, plug in the potential roots into x of the equation until one of them ends with a result of 0:
Since we know that 3 is a root, this means that one of the factors is (x - 3). Now that we know one of the roots, we are going to use synthetic division to divide the polynomial. To set it up, place the root of the divisor, in this case 3 from x - 3, on the left side and the coefficients of the original polynomial on the right side as such:
- 3 | 1 - 10 + 25 - 40 + 84
- _________________
Firstly, drop the 1:
- 3 | 1 - 10 + 25 - 40 + 84
- ↓
- _________________
- 1
Next, multiply 3 and 1, then add the product with -10:
- 3 | 1 - 10 + 25 - 40 + 84
- ↓ + 3
- _________________
- 1 - 7
Next, multiply 3 and -7, then add the product with 25:
- 3 | 1 - 10 + 25 - 40 + 84
- ↓ + 3 - 21
- _________________
- 1 - 7 + 4
Next, multiply 3 and 4, then add the product with -40:
- 3 | 1 - 10 + 25 - 40 + 84
- ↓ + 3 - 21 + 12
- _________________
- 1 - 7 + 4 - 28
Lastly, multiply -28 and 3, then add the product with 84:
- 3 | 1 - 10 + 25 - 40 + 84
- ↓ + 3 - 21 + 12 - 84
- _________________
- 1 - 7 + 4 - 28 + 0
Now our synthetic division is complete. Now since the degree of the original polynomial is 4, this means our quotient has a degree of 3 and follows the format . In this case, our quotient is
.
So right now, our equation looks like this:
However, our second factor can be further simplified. For the second factor, I will be factoring by grouping. So factor x³ - 7x² and 4x - 28 separately. Make sure that they have the same quantity inside the parentheses:
Now it can be rewritten as:
Since the polynomial cannot be further simplified, your answer is:
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The equation is
To find:
The slope intercept form of the given equation.
Solution:
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Where, m is slope and b is the y-intercept.
We have,
We need to isolate y. Add 10x on both sides.
Divide both sides by 20.
Therefore, the slope intercept form of the given equation is .
Even function:
A function is said to be even if its graph is symmetric with respect to the , that is:
Odd function:
A function is said to be odd if its graph is symmetric with respect to the origin, that is:
So let's analyze each question for each type of functions using examples of polynomial functions. Thus:
FOR EVEN FUNCTIONS:
1. When becomes
1.1 Effects on the y-intercept
We need to find out the effects on the y-intercept when shifting the function into:
We know that the graph intersects the y-axis when , therefore:
So:
So the y-intercept of is one unit less than the y-intercept of
1.2. Effects on the regions where the graph is increasing and decreasing
Given that you are shifting the graph downward on the y-axis, there is no any effect on the intervals of the domain. The function increases and decreases in the same intervals of
1.3 The end behavior when the following changes are made.
The function is shifted one unit downward, so each point of has the same x-coordinate but the output is one unit less than the output of . Thus, each point will be sketched as:
FOR ODD FUNCTIONS:
2. When becomes
2.1 Effects on the y-intercept
In this case happens the same as in the previous case. The new y-intercept is one unit less. So the graph is shifted one unit downward again.
An example is shown in Figure 1. The graph in blue is the function:
and the function in red is:
So you can see that:
2.2. Effects on the regions where the graph is increasing and decreasing
The effects are the same just as in the previous case. So the new function increases and decreases in the same intervals of
In Figure 1 you can see that both functions increase at:
and decrease at:
2.3 The end behavior when the following changes are made.
It happens the same, the output is one unit less than the output of . So, you can write the points just as they were written before.
So you can realize this concept by taking a point with the same x-coordinate of both graphs in Figure 1.
FOR EVEN FUNCTIONS:
3. When becomes
3.1 Effects on the y-intercept
We need to find out the effects on the y-intercept when shifting the function into:
As we know, the graph intersects the y-axis when , therefore:
And:
So the new y-intercept is the negative of the previous intercept shifted one unit upward.
3.2. Effects on the regions where the graph is increasing and decreasing
In the intervals when the function increases, the function decreases. On the other hand, in the intervals when the function decreases, the function increases.
3.3 The end behavior when the following changes are made.
Each point of the function has the same x-coordinate just as the function and the y-coordinate is the negative of the previous coordinate shifted one unit upward, that is:
FOR ODD FUNCTIONS:
4. When becomes
4.1 Effects on the y-intercept
In this case happens the same as in the previous case. The new y-intercept is the negative of the previous intercept shifted one unit upward.
4.2. Effects on the regions where the graph is increasing and decreasing
In this case it happens the same. So in the intervals when the function increases, the function decreases. On the other hand, in the intervals when the function decreases, the function increases.
4.3 The end behavior when the following changes are made.
Similarly, each point of the function has the same x-coordinate just as the function and the y-coordinate is the negative of the previous coordinate shifted one unit upward.
