State true or false: The polynomial

x
4

+4
x
2

+6
has no zero.

Question

State true or false: The polynomial

x
    4
   
   +4
    x
    2
   
   +6
  has no zero.

Accepted Answer

It is given that the polynomial

x
    4
   
   +4
    x
    2
   
   +6
  has no zeros. And we need to check it’s true or not.The zeros of a polynomial are such values of the variable at which the graph of a polynomial intersects the x-axis.So, we will plot the graph of the given function by setting it equal to y, a variable which represents the values on y-axis. So, let

y=
    x
    4
   
   +4
    x
    2
   
   +6
 . We have the table of values representing the ordered pairs which lie on the graph of the given polynomial as follows,

x

y=
    x
    4
   
   +4
    x
    2
   
   +6

(x,y)
 0

y=
      (0)
      4
     
     +4
      (0)
      2
     
     +6

y=0+0+6

y=6

(0,6)
 1

y=
      (1)
      4
     
     +4
      (1)
      2
     
     +6

y=1+4+6

y=11

(1,11)
 –1

y=
      (−1)
      4
     
     +4
      (−1)
      2
     
     +6

y=1+4+6

y=11

(−1,11)
 2

y=
      (2)
      4
     
     +4
      (2)
      2
     
     +6

y=16+16+6

y=38

(2,38)
 –2

y=
      (−2)
      4
     
     +4
      (−2)
      2
     
     +6

y=16+16+6

y=38

(−2,38)
 Now, plotting the graph using the ordered pairs as obtained above in the table, we get the graph as follows,So, in the above graph, we can observe that the curve which represents the polynomial does not intersect the x-axis at any point. Therefore, we conclude that the polynomial

x
    4
   
   +4
    x
    2
   
   +6
  has no zeros.Hence, the statement is true.So, the correct option is 1.

$x$	$y = x 4 + 4 x 2 + 6$	$(x, y)$
0	$y = (0) 4 + 4 (0) 2 + 6 y = 0 + 0 + 6 y = 6$	$(0, 6)$
1	$y = (1) 4 + 4 (1) 2 + 6 y = 1 + 4 + 6 y = 11$	$(1, 11)$
–1	$y = (− 1) 4 + 4 (− 1) 2 + 6 y = 1 + 4 + 6 y = 11$	$(− 1, 11)$
2	$y = (2) 4 + 4 (2) 2 + 6 y = 16 + 16 + 6 y = 38$	$(2, 38)$
–2	$y = (− 2) 4 + 4 (− 2) 2 + 6 y = 16 + 16 + 6 y = 38$	$(− 2, 38)$

State true or false: The polynomial $x 4 + 4 x 2 + 6$ has no zero.

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State true or false: The polynomial x 4 +4 x 2 +6 has no zero.