Question

If p(x) = 3x3 - 5x2 - x 2 then using synthetic substitution p(−1/3 ) =

Answer 1

If
`$P(x)=3 x^3-5 x^2-x+2$` then using synthetic substitution
`\( P\left((-1)/(3)\right) \approx 1.67 \)`.

To evaluate
`\( P\left((-1)/(3)\right) \)` for the polynomial
`\( P(x) = 3x^3 - 5x^2 - x + 2 \)` using synthetic substitution, we follow these steps:

1. Set Up the Synthetic Division Table: Write the coefficients of the polynomial in descending order of their powers. For
`\( P(x) \)`, the coefficients are
`\( 3, -5, -1, \)` and
`\( 2 \)`.

2. Write the Number to Substitute: This is
`\( (-1)/(3) \)`. We'll use this for the synthetic substitution.

3. Perform Synthetic Division:

• Bring down the first coefficient (3) as is.
• Multiply
  `\( (-1)/(3) \)` by the number just written down (3), and write the result under the next coefficient (-5).
• Add the numbers in this column and write the result beneath them.
• Repeat this process for each coefficient.

4. Find the Result: The final number in the bottom row is
`\( P\left((-1)/(3)\right) \)`.

Let's perform these steps.

The step-by-step synthetic substitution process for evaluating
`\( P\left((-1)/(3)\right) \)` is as follows:

1. Begin with the coefficients of
`\( P(x) \): \( [3, -5, -1, 2] \)`.

2. Perform the synthetic division steps:

• Bring down the first coefficient:
  `\( 3 \)`.
• Multiply
  `\( (-1)/(3) \)` by 3 and add to -5:
  `\( (-1)/(3) * 3 = -1, \)` then
  `\( -5 + (-1) = -6 \)`.
• Multiply
  `\( (-1)/(3) \)` by -6 and add to -1:
  `\( (-1)/(3) * -6 = 2, \)` then
  `\( -1 + 2 = 1 \)`.
• Multiply
  `\( (-1)/(3) \)` by 1 and add to 2:
  `\( (-1)/(3) * 1 = -(1)/(3), \)` then
  `\( 2 - (1)/(3) = (5)/(3) \)` or approximately
  `\( 1.67 \)`.

Therefore, The answer is 1.67.

Answer 2

The value of p(x) at x = -1/3 is 1/9.

What is synthetic substitution?

Synthetic substitution is a method for evaluating a polynomial at a specific value without long division. Given a polynomial and a value, it involves substituting the value into the equation to find the value of the equation.

Given

`p(x) = {3x}^(3) - {5x}^(2) - x </p><p>To find </p><p>[tex]p( ( - 1)/(3) )`

Substitute

`x = ( - 1)/(3)`

into p(x)

`p( ( - 1)/(3)) = {3 (( - 1)/(3) )}^(3) - {5( ( - 1)/(3)) }^(2) - (( - 1)/(3))`

`= (( - 3)/(27)) - ( 5)/(9) + (1)/(3)`

`= ( - 1)/(9) - (1)/(9) + (1)/(3)`

`= ( - 1 - 1 + 3 )/(9)`

`= (1)/(9)`

Therefore, the value of p(x) at x = -1/3 is 1/9.