All the real zeros of the given polynomial are integers. Find the zeros, and write the polynomial in factored form. P(x)=x3+3x2−4

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All the real zeros of the given polynomial are integers. Find the zeros, and write the polynomial in factored form.  P(x)=x3+3x2−4

jean2098 · Accepted Answer

Step 1  Given: All the real zeros of the given polynomial are integers.  Find the zeros, and write the polynomial in factored form.  P(x)=x3+3x2−4  Step 2  To Find the zeros, and write the polynomial in factored form:  P(x)=x3+3x2−4  =(x−1)(x+2)2  =(x-1)(x+2)(x+2)  Therefore, The zeros of P(x) is P(x)=0  Hence, (x-1)(x+2)(x+2)=0  ⇒x=1,−2,−2

Lynne Trussell · Accepted Answer

Consider the polynomial function in the textbook.  P(x)=x3+3x2−4  The objective of the questions is to find the zeros and write the polynomial in factored form.  Consider P(x)=x3+3x2−4,  The leading coefficient is 1 and the constant term is -4, any rational zero must be a divisior of the constant term -4.  So, the possible rational zeros are ±1,±2 and ±4  Test each of these possibilities.  Let x=1  P(x)=x3+3x2−4  P(1)=(1)3+3(1)2−4  =1+3-4  =0  Now test for x=-1  P(x)=x3+3x2−4  P(−1)=(−1)3+3(−1)−4  =-1+3-4  =-2  Similarly, for x=2  P(x)=x3+3x2−4  P(2)=(2)3+3(2)2−4  =8+12-4  =16  Similarly, for x=-2  P(x)=x3+3x2−4  P(−2)=(−2)3+3(−2)2−4  =-8+12-4  =0  Similarly, for x=4  P(x)=x3+3x2−4  P(4)=43+3(4)2−4  =64+48-4  =108  Similarly, for x=-4  P(x)=x3+3x2−4  P(−4)=(−4)3+3(−4)2−4  =-64-48-4  =-116  Hence, real zeros of P(x)=x3+3x2−4 are 1 and -2.  Now find the factored form of P(x)=x3+3x2−4,  P(x)=x3+3x2−4  P(x)=(x+2)2(x−1)  Hence, the factored form of P(x)=x3+3x2−4 is (x+2)2(x−1).

All the real zeros of the given polynomial are integers.

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