Brandy is asked to sketch a graph of, m(x) = (2x + 5)(x - 3)^3(8x + 1)^5. How many turning points should her sketch include?
The number of turning points in the graph of a polynomial function is equal to the number of real roots of its derivative.
First, let's find the derivative of m(x) using the product rule and the chain rule:
m'(x) = (2x + 5)(3(x - 3)^2)(8x + 1)^5 + (2(x - 3)^3)(8x + 1)^5 + (2x + 5)(x - 3)^3(5(8x + 1)^4)(8)
Simplifying this expression:
m'(x) = (3(2x + 5)(x - 3)^2)(8x + 1)^5 + (2(x - 3)^3)(8x + 1)^5 + (10(2x + 5)(x - 3)^3)(8x + 1)^4
Now, to find the number of real roots, we need to determine the number of sign changes in the coefficients of m'(x).
Let's examine the signs of each term of m'(x) as x approaches positive and negative infinity.
As x approaches positive infinity:
- The first term (3(2x + 5)(x - 3)^2)(8x + 1)^5) is positive.
- The second term (2(x - 3)^3)(8x + 1)^5) is negative.
- The third term (10(2x + 5)(x - 3)^3)(8x + 1)^4) is positive.
As x approaches negative infinity:
- The first term (3(2x + 5)(x - 3)^2)(8x + 1)^5) is negative.
- The second term (2(x - 3)^3)(8x + 1)^5) is negative.
- The third term (10(2x + 5)(x - 3)^3)(8x + 1)^4) is positive.
There is one sign change from positive to negative and one sign change from negative to positive. Therefore, the number of real roots of m'(x) is 2.
So, the graph of m(x) should have 2 turning points.
To determine the number of turning points on the graph of the function m(x), we need to examine the degree of the polynomial within the parentheses of each factor.
The function m(x) = (2x + 5)(x - 3)^3(8x + 1)^5 can be broken down into three factors:
1. (2x + 5): This factor is a linear function with a degree of 1, so it does not contribute any turning points.
2. (x - 3)^3: This factor is a cubic function with a degree of 3. Cubic functions can have either 1 or 3 turning points.
3. (8x + 1)^5: This factor is a quintic function with a degree of 5. Quintic functions can have either 2, 3, or 4 turning points.
To find the total number of turning points, we need to consider the product of the possible number of turning points in each factor:
- The first factor has 0 turning points.
- The second factor could have either 1 or 3 turning points.
- The third factor could have either 2, 3, or 4 turning points.
Taking all the possible combinations into account, the total number of turning points should be either 0, 1, 3, 2, 4, or 6.
To determine the number of turning points in the graph of the given function m(x), we need to look at the powers of the factors involved.
In the equation m(x) = (2x + 5)(x - 3)^3(8x + 1)^5, each factor has its own effect on the graph.
The factor (2x + 5) is a linear factor with a power of 1, which means it will contribute to a single turning point.
The factor (x - 3)^3 is a cubic factor with a power of 3. Cubic functions can have a maximum of two turning points, so this factor might contribute up to two turning points.
Lastly, the factor (8x + 1)^5 is a quintic factor with a power of 5. Quintic functions can have a maximum of four turning points, so this factor might contribute up to four turning points.
To find the total number of turning points, we need to consider the maximum number of turning points contributed by each factor: 1 (linear) + 2 (cubic) + 4 (quintic) = 7 turning points.
Therefore, Brandy's sketch should include at most 7 turning points.