Question

find all zeroes of the polynomial x4+x3-14x2-2x+24 if two zeroes are root 2 and -root 2​

Answer 1

Please see the attachment

Answer 2

Answer:

The other two zeroes will be at 3 and -4

Step-by-step explanation:

• The polynomial is, $x^4+ x^3 -16x^{2} -2x + 24$
• Factorization:  $x^4+ x^3 -16x^{2} -2x + 24$

                             =  $x^4 - 2x^2 + x^3 -2x^2 -12x^{2}+ 24$

                             = x²(x²-2) + x(x²-2) -12(x²-2)

                             = (x²-2)(x² + x -12)

• So here one factor is (x²-2).
• To make the polynomial zero, either (x²-2)= 0

                                                           or, (x² + x -12) = 0 or both can be zero

• When  (x²-2)= 0 then (x + √2)(x - √2) = 0

                                           x = √2 or -√2

• When (x² + x -12) = 0

                  x² +4x - 3x - 12 = 0

                  (x + 4)(x - 3) = 0

                  x = -4 or 3

∴ All zeros of the polynomial will be √2, -√2, -4 and 3