High School

Let [tex]P(x) = 5x^7 - 34x^6 + 10x^5 - 109x^4 - 60x^3 - 69x^2 + 12x - 125[/tex].

(a) Calculate [tex]P(7)[/tex] by using synthetic division.
[tex]P(7) =[/tex]

(b) Calculate [tex]P(7)[/tex] by substituting [tex]x = 7[/tex] into the polynomial and evaluating directly.

Answer :

To calculate P(7) for the polynomial P(x), synthetic division requires writing out the coefficients and systematically adding and multiplying by 7. Direct substitution simply involves replacing x with 7 in the polynomial and calculating the result according to the order of operations.

To calculate P(7) using synthetic division, we will follow these steps:

Write down the coefficients of P(x): 5, -34, 10, -109, -60, -69, 12, -125.

Place the number 7 outside the division symbol.

Bring down the first coefficient (5) and multiply it by 7, placing the result under the second coefficient (-34).

Add down the column and continue this process of multiplying by 7 and adding down the column until the last number is reached.

The final number is the value of P(7).

To calculate P(7) by direct substitution:

Substitute x with 7 in the polynomial P(x).

Follow the order of operations to evaluate the expression, starting with raising 7 to the appropriate powers and then applying multiplication, subtraction, and addition.

The final result will yield the value of P(7).