List all possible or potential rational zeros for the polynomial below find all real zeros of the polynomial below and factor completely over the real numbers.
f(x) x^4 + 10x^3 + 12x^2 - 58x + 35
Someone anyone?????
if any factors exist then, then some of ±1, ±5, ±7 must be zeros of the function
it is easy to see that x=1 works
f(1) = 1+10+12-58+35 = 0
by synthetic division,
x^4 + 10x^3 + 12x^2 - 58x + 35 = (x-1)(x^3 + 11x^2 + 23x - 35)
mmmh, looks like x=1 is also a zero of the cubic facor
another sysnthetic division left me with
x^4 + 10x^3 + 12x^2 - 58x + 35 =(x-1)(x-1)(x^2 + 12x + 35)
and that quadratic obviously factors again, so we have
x^4 + 10x^3 + 12x^2 - 58x + 35 =(x-1)^2 (x+5)(x+7)
Thank you!
To find the potential rational zeros of the polynomial f(x) = x^4 + 10x^3 + 12x^2 - 58x + 35, you can use the Rational Root Theorem.
1. List all possible factors of the constant term: ±1, ±5, ±7, ±35.
2. List all possible factors of the leading coefficient: ±1.
The potential rational zeros are the ratios of the factors of the constant term to the factors of the leading coefficient:
±1/1, ±5/1, ±7/1, ±35/1.
These are the potential rational zeros of the polynomial.
To find the real zeros and factor completely over the real numbers, you can use either synthetic division or a graphical method to find the zeros. As an AI text-based assistant, I cannot provide a graphical method, so I'll demonstrate how to use synthetic division to find the real zeros.
Step 1: Choosing a potential rational zero
Let's start with the potential rational zero x = 1.
Step 2: Synthetic division
Perform synthetic division with x = 1:
1 | 1 10 12 -58 35
| 1 11 23 -35
_________________________
1 11 23 -35 0
The result of the synthetic division is 1x^3 + 11x^2 + 23x - 35, with a remainder of 0.
Step 3: Next potential rational zero
Since the remainder is 0, x = 1 is a zero of the polynomial.
Step 4: Factoring
The division result, 1x^3 + 11x^2 + 23x - 35, can be factored to (x - 1)(x^2 + 12x + 35).
Now, let's factor the quadratic polynomial in the parentheses:
x^2 + 12x + 35 = (x + 5)(x + 7)
So the completely factored form of the original polynomial f(x) is:
f(x) = (x - 1)(x + 5)(x + 7)
The real zeros of the polynomial are x = 1, x = -5, and x = -7.
To find the possible rational zeros of a polynomial, we can use the Rational Root Theorem. According to the theorem, any rational zero of a polynomial can be expressed as a fraction of two integers, where the numerator divides the constant term and the denominator divides the leading coefficient.
For the polynomial f(x) = x^4 + 10x^3 + 12x^2 - 58x + 35, the constant term is 35 and the leading coefficient is 1. Therefore, the possible rational zeros can be obtained by considering the factors of the constant term 35 (±1, ±5, ±7, ±35) divided by the factors of the leading coefficient 1 (±1).
The possible rational zeros for the given polynomial are:
±1, ±5, ±7, ±35
To find the real zeros, we can use techniques such as factoring, synthetic division, or numerical methods. In this case, the polynomial is a quartic polynomial (degree 4), which means there might not be a straightforward way to significantly simplify or factor it. Therefore, we can use numerical methods like the Rational Root Theorem or a graphing calculator to find the approximate values of the real zeros.
To factor the polynomial completely over the real numbers, we start by finding any real zeros using the possible rational zeros we obtained earlier. We can use synthetic division or a calculator to evaluate the polynomial at these possible zeros and determine if they are indeed zeros.
After performing the calculations, you will find that some of the possible rational zeros are indeed zeros. For example, x = -1 is a zero of the polynomial. Dividing the polynomial by (x + 1) using long division or synthetic division will result in a quotient of x^3 + 9x^2 + 3x - 35.
Now, we can try finding the remaining zeros of the quotient polynomial x^3 + 9x^2 + 3x - 35. Since it is a cubic polynomial, we can apply the Rational Root Theorem again to find the possible rational zeros. The process is similar to what we described earlier.
By carrying out the calculations, you will find that x = -1, -5, and 7 are the real zeros of the cubic polynomial. Once again, you can divide the polynomial by each zero to obtain a new quotient.
Finally, we will be left with a quadratic polynomial after finding all the zeros. By solving this quadratic equation, we can determine the remaining real zeros.