๐ Understanding Synthetic Division
Synthetic division is a simplified method for dividing a polynomial by a linear factor of the form $x - c$. It's a faster alternative to long division, especially when dealing with simpler divisors.
๐ ๏ธ Materials Needed
- ๐ Paper or notebook
- โ๏ธ Pencil or pen
- ๐ฅ๏ธ Calculator (optional)
๐ฅ Warm-up (5 mins)
Before diving into synthetic division, make sure you're comfortable with:
- โ Basic arithmetic operations (addition, subtraction, multiplication, division)
- ๐ค Identifying coefficients and constants in polynomials
- ๐งฎ Understanding polynomial terminology (degree, leading coefficient, etc.)
๐งโ๐ซ Main Instruction: Step-by-Step Guide
Let's break down the process with an example: Divide $x^3 - 4x^2 + 6x - 4$ by $x - 2$.
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โ๏ธ Step 1: Set up the Synthetic Division
Write down the coefficients of the polynomial and the constant term 'c' from the divisor (x - c). In this case, c = 2.
Coefficients: 1, -4, 6, -4
Setup:
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โฌ๏ธ Step 2: Bring Down the First Coefficient
Bring down the first coefficient (1) below the line.
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โ๏ธ Step 3: Multiply and Add
Multiply the 'brought down' number (1) by 'c' (2), and write the result under the next coefficient (-4).
1 * 2 = 2
Add -4 and 2: -4 + 2 = -2
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๐ Step 4: Repeat
Repeat the multiplication and addition steps for the remaining coefficients.
-2 * 2 = -4
6 + (-4) = 2
| 2 |
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1 |
-4 |
6 |
-4 |
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| |
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2 |
-4 |
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1 |
-2 |
2 |
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2 * 2 = 4
-4 + 4 = 0
| 2 |
| |
1 |
-4 |
6 |
-4 |
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| |
|
2 |
-4 |
4 |
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1 |
-2 |
2 |
0 |
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๐ง Step 5: Interpret the Result
The last number (0) is the remainder. The other numbers (1, -2, 2) are the coefficients of the quotient, which is one degree lower than the original polynomial.
Quotient: $x^2 - 2x + 2$
Remainder: 0
Therefore, $(x^3 - 4x^2 + 6x - 4) / (x - 2) = x^2 - 2x + 2$
โ๏ธ Practice Problems
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โ Problem 1
Divide $2x^3 - 5x^2 + 3x + 4$ by $x - 1$.
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โ Problem 2
Divide $x^4 + 2x^3 - x + 5$ by $x + 2$.
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โ Problem 3
Divide $3x^3 - 7x^2 - 8x + 5$ by $x - 3$.
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โ Problem 4
Divide $x^3 + 8$ by $x + 2$.
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โ Problem 5
Divide $2x^4 - 5x^3 + x - 7$ by $x - 2$.
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โ Problem 6
Divide $x^4 - 16$ by $x - 2$.
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โ Problem 7
Divide $4x^3 + 2x^2 - x + 9$ by $x + 1$.
๐ Answer Key
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โ
Answer 1
Quotient: $2x^2 - 3x$; Remainder: $4 + 3 = 7$
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โ
Answer 2
Quotient: $x^3 - x + 1$; Remainder: $5-2 = 3$
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โ
Answer 3
Quotient: $3x^2 + 2x - 2$; Remainder: $5-6 = -1$
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โ
Answer 4
Quotient: $x^2 - 2x + 4$; Remainder: $8 - 8 = 0$
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โ
Answer 5
Quotient: $2x^3 - x - 2$; Remainder: $-7 -4 = -11$
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โ
Answer 6
Quotient: $x^3 + 2x^2 + 4x + 8$; Remainder: $16-16 = 0$
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โ
Answer 7
Quotient: $4x^2 - 2x + 1$; Remainder: $9-1 = 8$