Topic Polynomial ane Rational Functions Section Zeros of Pol

Name: Math&141 Daily HW 15 Due Tuesday, August 9. Instructions: You should be able to do all these problems without using a calculator or a graphing device 1a. Find a degree four polynomial function p(x) such that 4 and 6 are the only zeros, and each has multiplicity two. Give your answer in factored form. b. For your function from part (a), find the value of ) 2a. Find a degree two polynomial function with real coefficients that has no real zeros. Give your answer in simplified, not factored, form. 2b. Find a degree four polynomial function with real coefficients that has no real zeros. Give your answer in simplified, not factored, form.

Solution

1a) Since the polynomial should have zeroes of 4 and 6 with multiplicity 2, we can have the polynomial as

p(x) = (x-4)²(x-6)²

p(x) is of degree 4 and have zeroes as 4 and 6.

1b) p(7) = (7-4)²(7-6)² = 9

2a) Consider the polynomial x²+1

p(x) =0 gives x = i, -i

Hence no real roots

p(x) = x²+1

2b) Fourth degree polynomial with no real zeroes will have 4 complex roots of which pairs of conjugate roots exist

Hence let us take 1+i, 1-i, i, -i as roots

Then polynomial would be (x-1-i)(x-1+i)(x-i)(x+i)

= {(x-1)²+1}(x²+1)

= (x²-2x+2)(x²+1)

= x⁴-2x³+3x²-2x+2

p(x) = x⁴-2x³+3x²-2x+2 , a fourth degree with no real zeroes.