Form a polynomial whose zero and degree are given .zeros:_3,multiplicity 1 ;_1,multiplicity 2 ; degree 3 ?

Type a polynomial with integer coefficients and a leading coefficient.(simplify your answer)

Question

Form a polynomial whose zero and degree are given .zeros:_3,multiplicity 1 ;_1,multiplicity 2 ; degree 3 ?

Type a polynomial with integer coefficients and a leading coefficient.(simplify your answer)

Elijah Allen · Accepted Answer

# x^3-5x^2+7x-3#.

Let #p(x)# be the reqd. polynomial. We are given that #p(x)# has a zero #1# with multiplicity #2#. In other words, this means that #(x-1)# occurs twice as a factor of #p(x)#. Likewise, #(x-3)# appears as a factor of #p(x)# once. We are also given that the degree of #p(x)# is #3#. So, #p(x)# can not have more than #3# linear factors. Altogether, #p(x)=k(x-1)^2(x-3)," where, constant "k in RR-{0}#. We may take #k=1#, and get the desired polynomial, #p(x)=(x-1)^2(x-3)=x^3-5x^2+7x-3#. Enjoy Maths.!

Sadie Ackerman · Answer

undefined A polynomial with the given zeros and multiplicities can be formed by multiplying the factors corresponding to each zero. The polynomial with zeros at -3 (multiplicity 1), -1 (multiplicity 2), and with a degree of 3 is:

$$ (x + 3)(x + 1)^2 $$

Expanding this expression yields:

$$ (x + 3)(x^2 + 2x + 1) $$

$$ x^3 + 2x^2 + x + 3x^2 + 6x + 3 $$

$$ x^3 + 5x^2 + 7x + 3 $$

So, the polynomial is $ x^3 + 5x^2 + 7x + 3 $.