Question

Find the value of the polynomial $p\left( x \right)=5x-4{{x}^{2}}+3$ at
[i] x= 0 [ii] x= -1 [iii] x= 2

Answer 1

Hint: In order to find the value of the polynomial p(x) at x= a, we calculate the value of each of the terms of f(x) at x= a and hence find the value of p(x) at x= a.

Complete step-by-step answer:
[i] x =0
Substituting x = 0 in the expression of p(x), we get
$5\left( 0 \right)-4{{\left( 0 \right)}^{2}}+3=0-0+3=3$
Hence the value of p(x) at x= 0 is 3
[ii] x = -1
Substituting x = 1 in the expression of p(x), we get
$5\left( -1 \right)-4{{\left( -1 \right)}^{2}}+3=-5-4+3=-6$
Hence the value of p(x) at x = -1 is -6
[iii] x = 2
Substituting x = 2 in the expression of p(x), we get
$5\left( 2 \right)-4{{\left( 2 \right)}^{2}}+3=10-16+3=-3$
Hence the value of p(x) at x= 2 is -3

Note: Alternative method: Synthetic division: Best method.
In this method, we start by writing coefficients of the polynomial in order from the highest degree to the constant term. If in between some degree terms are missing we set their coefficient as 0.
Hence $p\left( x \right)=5x-4{{x}^{2}}+3=-4{{x}^{2}}+5x+3$ will be written as

Now the point which has to be substituted(say x= 0) is written as follows

0 is placed below the first term

Now the terms under the same column are added. The sum is then multiplied with the root, and the product is written under the coefficient of the next term.
Hence, we have

Continuing in this way we have the following

Since the last sum is 3, we have $p\left( 0 \right)=3$. This method also tells you what the quotient will be when p(x) is divided by x-0. Here it will be $-4\left( x \right)+5=-4x+5$
Similarly creating tables for x = -1 and x=2 , we get

Since the last sum is -6, we have $p\left( -1 \right)=-6$

Since the last sum is -3, we have $p\left( 2 \right)=-3$