Find all rational zeros of the polynomial, and write the polynomial in factored form. P(x)=4x3+4x2−x−1

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Find all rational zeros of the polynomial, and write the polynomial in factored form.   P(x)=4x3+4x2−x−1

Pohanginah · Accepted Answer

Step 1  Zeros of the function are those values of x for which the value of the function is 0.  To find the zeros of the polynomial, first we need to factor it.  First rewrite the polynomial as  4x3−x+4x2−1 and then factor out x and 1 from the first two term and the last two terms respectively, then factor out (4x2−1).  4x3−x+4x2−1=x(4x2−1)+1(4x2−1)  =(4x2−1)(x+1)  Step 2  Use the identity (a2−b2)=(a−b)(a+b) to further factor the polynomial as follows:  (4x2−1)(x+1)=((2x)2−12)(x+1)  =(2x-1)(2x+1)(x+1)  Therefore, the factored form of the polynomial is (2x−1)(2x+1)(x+1).  Step 3  Equate (2x−1)(2x+1)(x+1) to 0 and solve for x to find the zeros of the polynomial.  (2x-1)(2x+1)(x+1)=0  2x-1=0  x=12  2x+1=0  x=−12  x+1=0  x=-1  Therefore, the zeros are 12,−12 and -1.

Find all rational zeros of the polynomial, and write the polynomial in factored form. P(x)=4x^{3}+4x^{2}-x-1

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