Can someone please help with this one? "write a polynomial equation of degree 4 with leading coefficient 1 that has roots at -2, -1, 3, and 4."?

4 Answers
Apr 23, 2018

#(x+2)(x+1)(x-3)(x-4)#

Explanation:

This would give #x^4# as your largest power, expand the brackets to write it as an equation.

Apr 23, 2018

#x^4-4x^3-7x^2+22x+24#

Explanation:

Expand #(x+2)(x+1)(x-3)(x-4)#

#(x+2)(x+1)=x^2+x+2x+2=x^2+3x+2#

#(x^2+3x+2)(x-3)=x^3-3x^2+3x^2-9x+2x-6#

=#x^3-7x-6#

#(x^3-7x-6)(x-4)=x^4-4x^3-7x^2+28x-6x+24#

=#x^4-4x^3-7x^2+22x+24#

Apr 23, 2018

The polynomial is #x^4-4x^3-7x^2+22x+24=0#

Explanation:

A polynomial equation of degree #4# with leading coefficient #a# and four roots #alpha,beta,gamma,delta# is

#a(x-alpha)(x-beta)(x-gamma)(x-delta)=0#

Hence, a polynomial equation of degree #4# with leading coefficient #1# and four roots #-2,-1,3,4# is

#1(x-(-2))(x-(-1))(x-3)(x-4)=0#

or #(x+2)(x+1)(x-3)(x-4)=0#

or #(x^2+3x+2)(x^2-7x+12)=0#

or #x^4-7x^3+3x^3+12x^2+2x^2-21x^2-14x+36x+24=0#

or #x^4-4x^3-7x^2+22x+24=0#

Apr 23, 2018

#f(x) = (x+2)(x+1)(x-3)(x-4)#

Explanation:

You can answer the question by looking at the requirements and fulfilling them step by step. A degree 4 polynomial will have #x^4# in it somewhere, and 4 will be the largest number x is raised to in the equation. For example, #f(x) = x^4 + x^3 + x^2 + x + 1# is an equation with degree 4, since 4 is the largest number #x# is raised to in the equation.

A leading coefficient of 1 means that the highest order term (#x^4# in this case) has the coefficient of 1. For example, #f(x) = 2x^4 + x# has a leading coefficient of 2. You can think of #f(x) = x^4# as having a "secret" 1 infront of the #x^4# term. (i.e. #f(x) = 1x^4#)

Lastly, having roots at -2, -1, 3, and 4 can be fulfilled by writing the polynomial in a way where it is easy to identify roots. A root of the equation is a value of #x# where #f(x) = 0#. So for the equation #f(x)=(x-5)# the root is 5. Again, the roots for the equation #f(x)=(x-2)(x-3)# are 2 and 3, since we have #f(2) = 0# and #f(3) = 0#.

So all put together we can choose the equation #f(x) = (x+2)(x+1)(x-3)(x-4)# which has roots at -2, -1, 3, and 4. We can expand the equation to #f(x) = x^4 - 4 x^3 - 7 x^2 + 22 x + 24# to double check that the equation has a leading coefficient of 1.