The process is similar to long division with numbers.
Steps for Polynomial Long Division
Arrange both polynomials in descending order of powers
Divide the leading term of the dividend by the leading term of the divisor
Multiply the result by the divisor
Subtract from the dividend
Bring down the next term
Repeat until the degree of the remainder is less than the degree of the divisor
Example Format
๐ Practice Problems
1Problem 1medium
โ Question:
Use synthetic division to divide f(x)=2x3 by .
Explain using:
โ ๏ธ Common Mistakes: Polynomial Division and Remainder Theorem
Avoid these 4 frequent errors
๐ Real-World Applications: Polynomial Division and Remainder Theorem
See how this math is used in the real world
๐ Worked Example: Related Rates โ Expanding Circle
Problem:
A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of 2 cm/s. How fast is the area of the circle increasing when the radius is 10 cm?
What is Polynomial Division and Remainder Theorem?โพ
Understanding polynomial long division, synthetic division, and the Remainder and Factor Theorems
How can I study Polynomial Division and Remainder Theorem effectively?โพ
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Polynomial Division and Remainder Theorem study guide free?โพ
Yes โ all study notes, flashcards, and practice problems for Polynomial Division and Remainder Theorem on Study Mondo are 100% free. No account is needed to access the content.
What course covers Polynomial Division and Remainder Theorem?โพ
Polynomial Division and Remainder Theorem is part of the AP Precalculus course on Study Mondo, specifically in the Polynomial and Rational Functions section. You can explore the full course for more related topics and practice resources.
divisordividendโ=quotient+divisorremainderโ
Synthetic Division
Synthetic division is a shortcut for dividing by linear factors of the form (xโc).
When to Use Synthetic Division
โ Divisor is (xโc) (linear with leading coefficient 1)
โ Cannot use for divisors like (2xโ3) or (x2+1)
Steps for Synthetic Division
Write c (from xโc) outside the box
Write coefficients of the dividend inside
Bring down the first coefficient
Multiply by c, add to next coefficient
Repeat across all coefficients
Last number is the remainder
Example Setup
Dividing f(x) by (xโc):
c & a_n & a_{n-1} & \cdots & a_0 \\
& & ca_n & \cdots & \\
\hline
& a_n & b_{n-1} & \cdots & r
\end{array}$$
The bottom row gives quotient coefficients and remainder $r$.
## The Remainder Theorem
**Remainder Theorem**: When a polynomial $f(x)$ is divided by $(x - c)$, the remainder is $f(c)$.
$$f(x) = (x - c) \cdot q(x) + r$$
where $r = f(c)$
### Why It Matters
- Quick way to find remainders without full division
- Just evaluate $f(c)$!
## The Factor Theorem
**Factor Theorem**: $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$.
In other words:
- If $f(c) = 0$, then $(x - c)$ divides evenly into $f(x)$
- If the remainder is 0, then $c$ is a zero of $f(x)$
### Applications
1. **Testing for factors**: Check if $f(c) = 0$
2. **Finding zeros**: If $(x - c)$ is a factor, then $c$ is a zero
3. **Factoring**: Use known zeros to write factored form
## Rational Zero Theorem
If $f(x) = a_nx^n + ... + a_0$ has integer coefficients, then any rational zero $\frac{p}{q}$ must have:
- $p$ divides the constant term $a_0$
- $q$ divides the leading coefficient $a_n$
This gives us a list of **possible** rational zeros to test.
## Complete Factorization Strategy
1. Use Rational Zero Theorem to list possible zeros
2. Test candidates using synthetic division or direct evaluation
3. Once you find a zero $c$, factor out $(x - c)$
4. Repeat on the quotient polynomial
5. Factor completely over the real numbers
โ
5x2+
x+
2
(xโ2)
๐ก Show Solution
Solution:
Step 1: Set up synthetic division with c=2.
The coefficients of f(x) are: 2,โ5,1,2
2 & 2 & -5 & 1 & 2 \\
& & 4 & -2 & -2 \\
\hline
& 2 & -1 & -1 & 0
\end{array}$$
Step 2: Perform the operations.
- Bring down 2
- Multiply: $2 \times 2 = 4$, add: $-5 + 4 = -1$
- Multiply: $2 \times (-1) = -2$, add: $1 + (-2) = -1$
- Multiply: $2 \times (-1) = -2$, add: $2 + (-2) = 0$
Step 3: Interpret the result.
The bottom row gives: quotient coefficients $2, -1, -1$ and remainder $0$.
**Quotient**: $2x^2 - x - 1$
**Remainder**: $0$
Therefore:
$$f(x) = (x - 2)(2x^2 - x - 1)$$
Since the remainder is 0, $(x - 2)$ is a factor and $x = 2$ is a zero of $f(x)$.
**Answer:** Quotient: $2x^2 - x - 1$, Remainder: $0$
2Problem 2easy
โ Question:
Use the Remainder Theorem to find the remainder when f(x)=x4โ3x3+2xโ5 is divided by (x+1).
๐ก Show Solution
Solution:
The Remainder Theorem states that the remainder when dividing f(x) by (xโc) is f(c).
Step 1: Identify .
3Problem 3hard
โ Question:
Find all rational zeros of f(x)=2x3โx2โ13xโ6 and factor completely.
๐ก Show Solution
Solution:
Step 1: List possible rational zeros using the Rational Zero Theorem.
Factors of constant term (-6): ยฑ1,ยฑ2,ยฑ3,ยฑ6
Factors of leading coefficient (2): ยฑ1,ยฑ2
Possible rational zeros:
4Problem 4medium
โ Question:
Use synthetic division to divide: (2xยณ - 5xยฒ + 3x - 7) รท (x - 2)
๐ก Show Solution
Step 1: Set up synthetic division with divisor x - 2 (use 2):
2 | 2 -5 3 -7
Step 2: Bring down the first coefficient:
2 | 2 -5 3 -7
โ
2
Are there practice problems for Polynomial Division and Remainder Theorem?โพ
Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.