Find only the rational zeros of the function. If there are none, state this. f(x)=x^5-3 x^4-3 x^

Find only the rational zeros of the function. If there are...

Key Concepts

Rational Root Theorem

The Rational Root Theorem provides a method for listing all possible rational candidates for zeros of a polynomial with integer coefficients. It states that any possible rational zero can be expressed in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This theorem is crucial for narrowing down the candidates when searching for rational solutions.

Polynomial Zeros

Polynomial zeros are the values of the variable that make the polynomial equal to zero. Finding these zeros is essential because they correspond to the points where the polynomial graph crosses the horizontal axis. In the context of polynomial functions, knowing the zeros facilitates further factorization and analysis of the function's behavior.

Synthetic Division

Synthetic division is a streamlined method of dividing a polynomial by a binomial of the form (x - c) and is particularly useful for testing potential rational zeros. It simplifies the division process and helps determine whether a candidate zero, identified via the Rational Root Theorem, is an actual zero by checking if the remainder is zero.

Factor Theorem

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) equals zero. This theorem links the concept of zeros and factors, allowing one to factorize polynomials once a zero is found, thereby reducing the polynomial's degree and simplifying the problem of finding additional zeros.

Transcript

00:01 In this question, a function is given as x -resied to the power 5, minus 3x raised to the power 4, minus 3x cube plus 9x square, minus 4x plus 12.

00:14 We need to find out the rational zeros of this function.

00:19 We can see the degree of this function is 5, so this function can have at most 5 zeros.

00:26 Now according to rational zeros theorem, any rational 0 of a function can, be of form p upon q and the possibilities for this p and q are the possibility for p is are the factors of the last term which is 12 so the possibilities for p are plus minus 1 plus minus 2 plus minus 3 plus minus 4 plus minus 6 and plus minus the possibility for q are the factors of the coefficient of the leading term.

01:19 The coefficient of the leading term is 1.

01:21 So the factors of 1 are plus and minus 1.

01:25 So possibilities for q are plus minus 1.

01:31 Now from here we can figure out the possibilities of p upon q.

01:38 So the possibilities for p upon q are plus minus 1 plus minus 2.

01:45 Plus minus 3 plus minus 4 plus minus 6 and plus minus 12 we can see there are so many possibilities for the ratio zeros of this function but there can be a very few now we will check one by one for all these zeros possible zeros of this function but we will terminate when we get enough now we can use substitution method to check these numbers whether these are zeros of this function or not, but we will use synthetic division as synthetic division is more beneficial because the question during that synthetic division can be used for factization of that function.

02:51 So for one, we will check using synthetic division.

02:58 The coefficients of this function are 1, minus 3, minus 3, 9, minus 4, and 12.

03:18 Now we will use this.

03:24 First 1 is 0, so 1 plus 0 is 1.

03:29 Now 1 multiplied 1 is 1.

03:32 So here it is 1, minus 3 plus 1 is minus 2.

03:36 Minus 2 multiplied by 1 is minus 2 here we get minus 5 and minus 5 multiplied by 1 is minus 5 from here we get 4 4 multiplied by 1 is 4 here it is 0 1 multiplied by 0 is 0 now 12 minus 12 plus 0 is 12 so the remainder is not 0 from here we get f of 1 is not 1 from here we get f of 1 is not equal to 0.

04:14 Hence, 1 is not a 0 of this function.

04:19 Now we will move forward to check 2 as a 0 of this function.

04:31 We will use 2 for combining whether it is 0 is not...

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