We will sometimes talk about "evaluating" a polynomial. Evaluating a polynomial simply means plugging in a particular value for the variable (or particular values for the variables, in the case of a multivariate polynomial) and finding out the total value of the expression.

Example: Find the value of x² +3x + 2 when x = 7.

Solution: 7² + 3(7) + 2 = 49 + 21 + 2 = 72.

Example: Find the value of 7x³ - x when x =2.

Solution: 7(2³) - 2 = 7(8) - 2 = 56 - 2 = 54.

Example: Find the value of 2x + 3y - xy, when x = 5 and y = - 1.

Solution: 2(5) +3(-1) - 5(-1) = 10 - 3 + 5 = 12.

Evaluating polynomials also gives rise to the concept of polynomial equations:

Example: Find a value of x for which the polynomial x² + 3x + 2 evaluates to 20.

Solution: We can turn this into a polynomial equation simply be writing x² + 3x + 2 = 20. In general, when confronted with a polynomial equation (unless it is of degree 1) our goal is to get everything on one side, and zero on the other: x² + 3x - 18 = 0. Thus, (x - 3)(x + 6) = 0, so x = 3 or x = -6.

1.

Evaluate the polynomial x² - 4 when x = 10

2.

Evaluate the polynomial x⁵ + 1 when x = 2

3.

Evaluate the polynomial x³ + 3x² + 3x + 1 when x = 10

4.

Find values of x for which the polynomial x² + 10x evaluates to 24.

5.

Find a value of x for which the polynomial x³ + x² and the polynomial x³ + 2x - 1 have the same value.