Find only the rational zeros of the function. If there are none, state this. f(x)=x^4+2 x^3-5 x^

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

Key Concepts

Polynomial Zeros

Polynomial zeros are the solutions to the equation f(x) = 0, where f(x) is a polynomial function. Identifying zeros is a fundamental concept in algebra as it reveals the x-intercepts of the polynomial's graph and is crucial in understanding its behavior and factorization.

Rational Root Theorem

The Rational Root Theorem is a principle used to identify all possible rational zeros of a polynomial function with integer coefficients. It states that any potential rational zero can be expressed as a fraction p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Factor Theorem

The Factor Theorem states that if a polynomial f(x) has a zero at x = r, then (x - r) is a factor of f(x). This concept is essential when confirming whether a candidate, suggested by the Rational Root Theorem, is indeed a zero by showing that the polynomial evaluates to zero at that candidate.

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a factor of the form (x - r). It is a useful technique for testing potential zeros, as it helps determine whether a candidate zero results in a remainder of zero, thereby confirming it as a valid zero.

Transcript

00:01 In this question, function f of x is given as x x x x to the power 4 plus 2 x q minus 5 x squared minus 4x plus 6.

00:11 We need to find out the rational zeros of this function.

00:15 As we see, the degree of this function is 4.

00:19 So at most 4 zeros are possible of this function.

00:25 Now according to the rational zero theorem, any rational 0 of a function must be of form p upon q where this p is a factor of this last term and q is a factor of coefficient of the first term so from here the possible values of p are plus minus one plus minus two plus minus three and plus minus six which are the factors of six and the possible values of q is only plus minus from here we can figure out the possible values of p upon q which is a possible zero of this function f of x.

01:24 So the possible values of p upon q are plus minus 1 plus minus 2 plus minus 3 plus minus 6.

01:44 Now we will check for these values whether these which are which of these is a zero of this function and which are not we can use substitution method but we will use synthetic division method as it is more useful perfectizing this function now we will check whether one is a zero of this function or not so one is here now coefficients of this function are 1 2 minus 5 minus 4 and 6 and now we have first of all there's 0 now 1 plus 0 is 1 multiply 1 by 1 is 1 now 2 plus 1 is 3 multiply 1 by 3 it is 3 now minus 5 plus 3 is minus 2 1 multiplied by minus 2 is minus 2 now 1 multiplied by minus 2 is so minus 4 minus 2 is minus 6 now 1 multiplied by minus 6 is minus 6 is 6 we have 6 minus 6 is 0 from here we see that remainder of this division is 0 so 1 is a 0 of this function so we can say that f of 1 is 0 hence 1 is a 0 of this function 1 is a 0 of x now we can factorize function f of x using the coefficient of this synthetic division.

03:52 So f of x can be factored as one of its factor will be x minus 1 as 1 is a 0 of function f of x.

04:01 So taking it out x minus 1 as a factor another factor will remain a one which found by these quotients.

04:11 So it is x q plus 3x square minus 2x minus 6 now we will check another possible value for the 0 of this factor only as we know if a value is 0 of this factor then it will also a 0 of f of x so we will check whether 2 is a 0 of this factor or not using synthetic division so using 2 for checking whether it is a 0 of this function or not we will use coefficients of this expression which are 1 3 minus 2 and minus 6 now here it is 0 1 plus 0 is 1 and 1 multiplied by 2 is 2 so 3 plus 2 is 5 now 5 multiplied by 2 is 10 minus 2 plus 10 is 8 now 8 multiplied by 2 is 16 minus 6 plus 16 is plus 10.

05:35 So from here we see that the remainder is not 0, as 2 is not a factor of either this expression nor it is a factor of f of x...

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