Synthetic Division

Cheatsheet Content

### Introduction Synthetic division is a shorthand method of dividing polynomials when the divisor is of the form $x - k$. It's a faster alternative to long division for specific cases. ### When to Use - **Divisor:** Must be a linear binomial of the form $x - k$. - Example: $x - 2$, $x + 3$ (which is $x - (-3)$) - **Dividend:** Can be any polynomial. - **Result:** Provides the quotient and the remainder of the polynomial division. ### Steps for Synthetic Division Let's divide $P(x) = ax^n + bx^{n-1} + ... + c$ by $x - k$. 1. **Identify $k$:** From the divisor $x - k$, $k$ is the constant term. If the divisor is $x + k'$, then $k = -k'$. 2. **Write Coefficients:** Write down the coefficients of the dividend polynomial $P(x)$ in order of descending powers. If any power is missing, use a zero as its coefficient. - Example: For $x^3 + 2x - 5$, coefficients are $1, 0, 2, -5$. 3. **Set Up:** Draw an upside-down division box. Put $k$ to the left. Write the coefficients of $P(x)$ to the right. ``` k | a b c d ... | -------------------- ``` 4. **Bring Down:** Bring the first coefficient (a) straight down below the line. ``` k | a b c d ... | -------------------- a ``` 5. **Multiply:** Multiply the number you just brought down (a) by $k$. Write the result under the next coefficient (b). ``` k | a b c d ... | ak -------------------- a ``` 6. **Add:** Add the numbers in the second column ($b + ak$). Write the sum below the line. ``` k | a b c d ... | ak -------------------- a (b+ak) ``` 7. **Repeat:** Continue multiplying the latest sum by $k$ and adding to the next coefficient until all coefficients have been processed. ``` k | a b c d | ak k(b+ak) k(c+k(b+ak)) --------------------------------- a (b+ak) (c+k(b+ak)) (d+k(c+k(b+ak))) ``` 8. **Interpret Results:** - The numbers below the line (except the last one) are the coefficients of the **quotient polynomial**. - The last number below the line is the **remainder**. - The degree of the quotient polynomial is one less than the degree of the dividend polynomial. If $P(x)$ divided by $x-k$ yields quotient $Q(x)$ and remainder $R$, then: $P(x) = (x-k)Q(x) + R$ Also, by the Remainder Theorem, $P(k) = R$. ### Example Divide $P(x) = x^3 - 4x^2 + 2x + 5$ by $x - 2$. 1. **Identify $k$**: $x - 2 \implies k = 2$. 2. **Coefficients**: $1, -4, 2, 5$. 3. **Set Up**: ``` 2 | 1 -4 2 5 | ------------------ ``` 4. **Bring Down**: ``` 2 | 1 -4 2 5 | ------------------ 1 ``` 5. **Multiply & Add (Column 2)**: $2 \times 1 = 2$. $-4 + 2 = -2$. ``` 2 | 1 -4 2 5 | 2 ------------------ 1 -2 ``` 6. **Multiply & Add (Column 3)**: $2 \times (-2) = -4$. $2 + (-4) = -2$. ``` 2 | 1 -4 2 5 | 2 -4 ------------------ 1 -2 -2 ``` 7. **Multiply & Add (Column 4)**: $2 \times (-2) = -4$. $5 + (-4) = 1$. ``` 2 | 1 -4 2 5 | 2 -4 -4 ------------------ 1 -2 -2 1 ``` 8. **Interpret Results**: - Quotient coefficients: $1, -2, -2$. This means $1x^2 - 2x - 2$. - Remainder: $1$. Therefore, $(x^3 - 4x^2 + 2x + 5) \div (x - 2) = x^2 - 2x - 2 + \frac{1}{x - 2}$. And $P(2) = 1$. ### Special Cases - **Missing Terms:** Always use zero for missing powers of $x$. - Example: $x^4 - 3x^2 + 1$ by $x - 1 \implies$ coefficients $1, 0, -3, 0, 1$. - **Divisor with Leading Coefficient:** If the divisor is $ax - k$, first divide the entire polynomial by $a$ before performing synthetic division with $k/a$. - Example: To divide $2x^2 + 5x + 3$ by $2x + 1$: 1. Rewrite $2x + 1$ as $2(x + 1/2)$, so $k = -1/2$. 2. Divide the polynomial by $2$: $x^2 + \frac{5}{2}x + \frac{3}{2}$. 3. Perform synthetic division with $k = -1/2$ on $x^2 + \frac{5}{2}x + \frac{3}{2}$. 4. The result will be the quotient.