Write a polynomial f (x) that meets the given conditions and has integer coefficients.&nbsp;Lowest degree polynomial with f(0) = -108 and zeros of -2 (multiplicity 2) and 3 (multiplicity 2)

Question

Write a polynomial f (x) that meets the given conditions and has integer coefficients.&amp;nbsp;Lowest degree polynomial with f(0) = -108 and zeros of -2 (multiplicity 2) and 3 (multiplicity 2)

William Appel · Accepted Answer

Step 1&amp;nbsp;Given:&amp;nbsp;Polynomial of lowest degree with zeros of -2 (multiplicity 2) and 3(multiplicity 2)&amp;nbsp;and with f(0)=-108.&amp;nbsp;Step 2&amp;nbsp;The polynomial equation is provided by:⇒f(x)=A(x+2)2(x−2)3&amp;nbsp;When, x=0, f(0)=-108:&amp;nbsp;⇒−108=A(0+2)2(0−2)3&amp;nbsp;⇒−108=A(2)2(−2)3&amp;nbsp;⇒−108=A(4)(−8)&amp;nbsp;⇒−108=−32A&amp;nbsp;Step 3&amp;nbsp;⇒−108−32=A&amp;nbsp;⇒3.375=A&amp;nbsp;⇒A=3.375&amp;nbsp;⇒f(x)=3.375(x+2)2(x−2)3

Archie Jones · Accepted Answer

Step 1&amp;nbsp;To find a polynomial f (x) with integer coefficients that satisfies the requirementsStep 2&amp;nbsp;Polynomial of lowest-degree&amp;nbsp;with zeroes of -2 multiplicity 2...(i)&amp;nbsp;zeros of 3 multiplicity 3...(ii)&amp;nbsp;f(0)=-108&amp;nbsp;i*e,&amp;nbsp;(x+2)2=0&amp;nbsp;(x−3)3=0&amp;nbsp;or&amp;nbsp;f(x)=A(x+2)2(x−3)3&amp;nbsp;f(0)=A(4)(-27)&amp;nbsp;-108=A(-108)&amp;nbsp;A=1&amp;nbsp;∴f(x)=(x+2)2(x−3)3&amp;nbsp;Is the polynomial with degree of 5.

Vasquez · Accepted Answer

Solution: zeros of -2 (multiplicity 2) zeros of 3 (multiplicity 2) △f(0)=−108⇒f(x)=k(x+2)2(x−3)2f(0)=−108=k(4)(9)=36kk=−10836=−3⇒f(x)=−3(x+2)2(x−3)2

Write a polynomial f (x) that meets the given conditions

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