Question

If $ p\left( x \right) = {x^3} + 2{x^2} - 5x - 6 $ , find $ p\left( 2 \right),p\left( { - 1} \right),p\left( { - 3} \right) $ and $ p\left( 0 \right) $ . What can you say about zeros of $ p\left( x \right) $ .

Answer 1

Hint: Given expression is a cubic expression. So it will have three zeroes (solutions). So substitute 2, -1, -3 and 0 in the place of x of $ p\left( x \right) $ and find the value of the expressions in each case. If the expression value is equal to zero, then that particular value of x is the zero of the given cubic expression.

Complete step-by-step answer:
We are given to find the values of $ p\left( 2 \right),p\left( { - 1} \right),p\left( { - 3} \right) $ , $ p\left( 0 \right) $ and the zeroes of the expression $ p\left( x \right) = {x^3} + 2{x^2} - 5x - 6 $ .
For finding the value of $ p\left( 2 \right) $ , we are substituting 2 in the place of x in $ p\left( x \right) $
 $ \Rightarrow p\left( 2 \right) = {2^3} + 2\left( {{2^2}} \right) - 5\left( 2 \right) - 6 = 8 + 8 - 10 - 6 = 16 - 16 = 0 $
 For finding the value of $ p\left( { - 1} \right) $ , we are substituting -1 in the place of x in $ p\left( x \right) $
 $ \Rightarrow p\left( { - 1} \right) = {\left( { - 1} \right)^3} + 2\left( { - {1^2}} \right) - 5\left( { - 1} \right) - 6 = - 1 + 2 + 5 - 6 = 7 - 7 = 0 $
 For finding the value of $ p\left( { - 3} \right) $ , we are substituting -3 in the place of x in $ p\left( x \right) $
 $ \Rightarrow p\left( { - 3} \right) = {\left( { - 3} \right)^3} + 2\left( { - {3^2}} \right) - 5\left( { - 3} \right) - 6 = - 27 + 18 + 15 - 6 = 33 - 33 = 0 $
 For finding the value of $ p\left( 0 \right) $ , we are substituting 0 in the place of x in $ p\left( x \right) $
 $ \Rightarrow p\left( 0 \right) = {\left( 0 \right)^3} + 2\left( {{0^2}} \right) - 5\left( 0 \right) - 6 = 0 + 0 - 0 - 6 = - 6 $
As we can see, the values of $ p\left( 2 \right),p\left( { - 1} \right),p\left( { - 3} \right) $ are zero and the value of $ p\left( 0 \right) $ is not zero. This means that 2, -1 and -3 are the zeroes of the given cubic expression $ p\left( x \right) = {x^3} + 2{x^2} - 5x - 6 $ .

Note: No. of zeros (solutions) of an expression or an equation depends upon the highest degree of its variable. If the highest degree is 2, then that expression or equation will have exactly 2 solutions; if the highest degree is 4, then it will have exactly 4 solutions. Here the highest degree of x is 3 so the above expression had exactly 3 solutions. Substituting the value of x as zero will not give the expression value as zero always. So do not confuse that the number ‘0’ is not a zero (solution) of an equation always.