Answer:
total 3 roots,
How do I know?:
put this function in graphing calculator
And if you don't have a graphing calculator, then calculate it out bu yourself:
(x + 1)(x − 1)(x − 7) = 0
There are three roots for the given function H(x) = x³ - 7x² - x + 7. The given polynomial has 3 as the highest degree, then it has 3 roots.
Part(I): There are 3 roots for the given polynomial equation. Since it has the highest degree as 3, there will be three values that satisfy the equation.
They are,
H(x) = x³ - 7x² - x + 7
H(x) = (x - 7)(x - 1)(x + 1)
roots are 7, 1, and -1.
Part(II): Using Descarte's rule, finding the positive zeros of the given polynomial:
H(x) = x³ - 7x² - x + 7
Since x = +x
there are two changes in the sign for the polynomial. One change is at the coefficient of x³ to the coefficient of x² and the other change is at the coefficient of x to the constant.
So, there are two positive zeros and they are 7 and 1.
Part(III): Using Descarte's rule, finding the negative zeros of the given polynomial:
H(x) = x³ - 7x² - x + 7
Substituting x = -x
H(-x) = -x³ - 7x² + x + 7
Only one sign is changed in the polynomial. That is at the coefficient of x² to the coefficient of x.
So, there is only one negative zero and it is -1.
Part(IV): The roots of the given equation are -1, 1, and 7. There are no complex roots for this equation. All the roots are real.
Part(V): If x = -1 is one of the roots of the polynomial, solving the leftover by using the synthetic division method as follows:
-1 ) 1 -7 -1 7
0 -1 8 -7
__________
1 -8 7 0
So, the leftover quadratic equation is x² - 8x + 7.
Solving this equation,
consider x = 1,
1) 1 -8 7
0 1 -7
________
1 -7 0
So, the quotient is (x - 7)
Therefore, the equation H(x) = x³ - 7x² - x + 7 is written as H(x) = (x - 7) (x - 1) (x + 1).
Learn more about the roots of a polynomial here:
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