Question

How do you use synthetic division and Remainder theorem to find \[P\left( a \right)\] if \[P\left( x \right) = {x^3} - 8{x^2} + 5x - 7\] and \[a = 1\]

Answer 1

Hint: - Given a polynomial. We have to apply the synthetic division and the remainder theorem. First, we will write the coefficient of all terms and value of a in line. Then, bring down the first coefficient in dividend which will become the quotient. Then multiply the value of a by quotient and write the result in the next column. Add the numbers in the second column and write the result in the bottom of the row. Repeat the steps until we reach the end of the division.

Complete step by step answer:
Here the polynomial is, \[P\left( x \right) = {x^3} - 8{x^2} + 5x - 7\] and the divisor is \[a = 1\]

First, write the coefficients of each term in descending order and the divisor. Then, bring down the first coefficient that is 1.

Then, multiply 1 and 1, and write the result \[1 \times 1 = 1\] in the next column below the second coefficient.

Then, add the values \[ - 8 + 1 = - 7\] and write the result in the second column below.

Now, the next quotient is \[ - 7\] which is again multiplied by divisor 1. Then, the result \[ - 7 \times 1 = - 7\] is written in the third column below the coefficient 5.

Then, we will add \[ - 7\] and 5. The result \[ - 7 + 5 = - 2\] is written in the third column and row below the line.

Now, the next quotient is \[ - 2\] which is again multiplied by divisor 1. Then, the result \[ - 2 \times 1 = - 2\] is written in the fourth column below the coefficient \[ - 7\].

Then, we will add \[ - 2\] and \[ - 7\]. The result \[ - 2 - 7 = - 9\] is written in the fourth column and row below the line.

\[1)1{\text{ }} - 8{\text{ 5 }} - 7\]
           \[1\] \[ - 7\] \[ - 2\]
  \[\overline {1{\text{ }} - 7{\text{ }} - 2{\text{ }} - 9{\text{ }}} \]

Here, the last number is \[ - 9\] which means the remainder of the synthetic division is \[ - 9\].

Now, apply the remainder theorem, we get:

\[P\left( 1 \right) = - 9\]

Hence the remainder of division of \[P\left( x \right) = {x^3} - 8{x^2} + 5x - 7\] by \[a = 1\] is \[ - 9\].

Note: Please note that the remainder theorem states that the remainder is \[f\left( a \right)\] when a polynomial \[f\left( x \right)\] is divided by the divisor \[x - a\]. Also, make sure that 0 must be inserted for the missing terms and the coefficients of the polynomial written in the division box must be in decreasing order of degree.