Pre-Calculus 12
Unit 3 Polynomial Functions.
Name: ______________________
Lesson Notes: 3.2: The Remainder Theorem
Focus on:
•describing the relationship between polynomial long division and synthetic division
• dividing polynomials by binomials of the form x – a using long division or synthetic division
• explaining the relationship between the remainder when a polynomial is divided by a
binomial of the form x – a and the value of the polynomial at x = a
Investigate Polynomial Division
Examine the two long-division statements.
Determine a Remainder
Compare the values of each remainder from the long division to the value from substituting x = a
into the dividend. What do you notice?
Make a conjecture about how to determine a remainder without using division.
Example 1) a) Divide the polynomial P(x) = 5x3 + 10x – 13x2 – 9 by x – 2. Express the result in the
P ( x)
R
form
= Q(x) +
xa
xa
b) Identify any restrictions on the variable.
c) Write the corresponding statement that can be used to check the division.
d) Verify your answer.
Your Turn. a) Divide the polynomial P(x) = x4 – 2x3 + x2 – 3x + 4 by x – 1. Express the result in the
P ( x)
R
= Q(x) +
form
xa
xa
b) Identify any restrictions on the variable.
c) Verify your answer.
Example 2) The volume, V, of the nested boxes in the introduction to this section, in cubic cm, is given
by V(x) = x3 + 7x2 + 14x + 8. What are the possible dimensions of the boxes in terms of x if
the height, h, in cm, is x + 1?
Your Turn.
The volume of a rectangular prism is given by V(x) = x3 + 3x2 – 36x + 32. Determine possible measures
for w and h in terms of x if the length, l, is x – 4.
Synthetic division is an alternate process for dividing a polynomial by a binomial of the
form x – a. It allows you to calculate without writing variables and requires fewer calculations.
3
2
Example 3) a) Use synthetic division to divide 2x + 3x – 4x + 15 by x + 3.
b) Check the results using long division.
Your Turn. Use synthetic division to determine
x3  7 x 2  3x  4
.
x2
The remainder theorem states that when a polynomial in x, P(x), is divided by a binomial of
the form x – a, the remainder is P(a).
Example 4) a) Use the remainder theorem to determine the remainder when P(x) = x3 – 10x + 6 is
divided by x + 4.
b) Verify your answer using synthetic division.
Your Turn.
What is the remainder when 11x – 4x4 – 7 is divided by x – 3? Verify
your answer using either long or synthetic division.
Example 5) For each dividend, determine the value of k if the remainder is –2.
3
2
a) (2x – 5x – 4x + k) ÷ (x + 1)
3
2
b) (x – 4x + kx + 10) ÷ (x – 3)
3
2
Example 6) For what value of m will the polynomial P(x) = x + 6x + mx – 4 have the same
remainder when it is divided by x – 1 and x + 2?