Synthetic Division Polynomials worksheet with polynomial division problems.
Worksheet titled "What Has Three Feet But No Toes?" featuring eight synthetic division problems involving polynomials.
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Show Answer Key & Explanations
Step-by-step solution for: Synthetic Division Worksheet Activity {Dividing Polynomials Activity}
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Show Answer Key & Explanations
Step-by-step solution for: Synthetic Division Worksheet Activity {Dividing Polynomials Activity}
Here are the solutions to the synthetic division problems on your worksheet. I have worked through each one step-by-step to ensure accuracy.
1. $(x^2 + 7x - 10) \div (x - 8)$
* Setup: The divisor is $x - 8$, so we use positive $8$ on the outside. The coefficients are $1, 7, -10$.
* Step 1: Bring down the $1$.
* Step 2: Multiply $1 \times 8 = 8$. Add to next coefficient: $7 + 8 = 15$.
* Step 3: Multiply $15 \times 8 = 120$. Add to next coefficient: $-10 + 120 = 110$.
* Result: The quotient is $x + 15$ with a remainder of $110$.
2. $(x^2 + 10x + 15) \div (x + 5)$
* Setup: The divisor is $x + 5$, so we use negative $-5$ on the outside. Coefficients: $1, 10, 15$.
* Step 1: Bring down the $1$.
* Step 2: Multiply $1 \times -5 = -5$. Add to next coefficient: $10 + (-5) = 5$.
* Step 3: Multiply $5 \times -5 = -25$. Add to next coefficient: $15 + (-25) = -10$.
* Result: The quotient is $x + 5$ with a remainder of $-10$.
3. $(x^3 + 7x^2 + 14x + 3) \div (x + 2)$
* Setup: Divisor $x + 2$ means use $-2$. Coefficients: $1, 7, 14, 3$.
* Step 1: Bring down $1$.
* Step 2: $1 \times -2 = -2$. Add: $7 + (-2) = 5$.
* Step 3: $5 \times -2 = -10$. Add: $14 + (-10) = 4$.
* Step 4: $4 \times -2 = -8$. Add: $3 + (-8) = -5$.
* Result: The quotient is $x^2 + 5x + 4$ with a remainder of $-5$.
4. $(x^2 - 72) \div (x - 8)$
* Setup: Divisor $x - 8$ means use $8$. Note that the $x$ term is missing, so we must use a $0$ placeholder. Coefficients: $1, 0, -72$.
* Step 1: Bring down $1$.
* Step 2: $1 \times 8 = 8$. Add: $0 + 8 = 8$.
* Step 3: $8 \times 8 = 64$. Add: $-72 + 64 = -8$.
* Result: The quotient is $x + 8$ with a remainder of $-8$.
5. $(x^3 - 13x^2 + 49x + 15) \div (x - 7)$
* Setup: Divisor $x - 7$ means use $7$. Coefficients: $1, -13, 49, 15$.
* Step 1: Bring down $1$.
* Step 2: $1 \times 7 = 7$. Add: $-13 + 7 = -6$.
* Step 3: $-6 \times 7 = -42$. Add: $49 + (-42) = 7$.
* Step 4: $7 \times 7 = 49$. Add: $15 + 49 = 64$.
* Result: The quotient is $x^2 - 6x + 7$ with a remainder of $64$.
6. $(x^2 - 28) \div (x - 5)$
* Setup: Divisor $x - 5$ means use $5$. Missing $x$ term requires a $0$ placeholder. Coefficients: $1, 0, -28$.
* Step 1: Bring down $1$.
* Step 2: $1 \times 5 = 5$. Add: $0 + 5 = 5$.
* Step 3: $5 \times 5 = 25$. Add: $-28 + 25 = -3$.
* Result: The quotient is $x + 5$ with a remainder of $-3$.
7. $(x^3 + 30x - 18 - 4x^2) \div (x + 5)$
* Setup: First, rearrange into standard form: $x^3 - 4x^2 + 30x - 18$. Divisor $x + 5$ means use $-5$. Coefficients: $1, -4, 30, -18$.
* Step 1: Bring down $1$.
* Step 2: $1 \times -5 = -5$. Add: $-4 + (-5) = -9$.
* Step 3: $-9 \times -5 = 45$. Add: $30 + 45 = 75$.
* Step 4: $75 \times -5 = -375$. Add: $-18 + (-375) = -393$.
* Result: The quotient is $x^2 - 9x + 75$ with a remainder of $-393$.
8. $(8x + x^3 - 5x^2 + 4) \div (-1 + x)$
* Setup: Rearrange dividend: $x^3 - 5x^2 + 8x + 4$. Rearrange divisor: $x - 1$. Use $1$ on the outside. Coefficients: $1, -5, 8, 4$.
* Step 1: Bring down $1$.
* Step 2: $1 \times 1 = 1$. Add: $-5 + 1 = -4$.
* Step 3: $-4 \times 1 = -4$. Add: $8 + (-4) = 4$.
* Step 4: $4 \times 1 = 4$. Add: $4 + 4 = 8$.
* Result: The quotient is $x^2 - 4x + 4$ with a remainder of $8$.
Final Answer:
1. $x + 15, R 110$
2. $x + 5, R -10$
3. $x^2 + 5x + 4, R -5$
4. $x + 8, R -8$
5. $x^2 - 6x + 7, R 64$
6. $x + 5, R -3$
7. $x^2 - 9x + 75, R -393$
8. $x^2 - 4x + 4, R 8$
1. $(x^2 + 7x - 10) \div (x - 8)$
* Setup: The divisor is $x - 8$, so we use positive $8$ on the outside. The coefficients are $1, 7, -10$.
* Step 1: Bring down the $1$.
* Step 2: Multiply $1 \times 8 = 8$. Add to next coefficient: $7 + 8 = 15$.
* Step 3: Multiply $15 \times 8 = 120$. Add to next coefficient: $-10 + 120 = 110$.
* Result: The quotient is $x + 15$ with a remainder of $110$.
2. $(x^2 + 10x + 15) \div (x + 5)$
* Setup: The divisor is $x + 5$, so we use negative $-5$ on the outside. Coefficients: $1, 10, 15$.
* Step 1: Bring down the $1$.
* Step 2: Multiply $1 \times -5 = -5$. Add to next coefficient: $10 + (-5) = 5$.
* Step 3: Multiply $5 \times -5 = -25$. Add to next coefficient: $15 + (-25) = -10$.
* Result: The quotient is $x + 5$ with a remainder of $-10$.
3. $(x^3 + 7x^2 + 14x + 3) \div (x + 2)$
* Setup: Divisor $x + 2$ means use $-2$. Coefficients: $1, 7, 14, 3$.
* Step 1: Bring down $1$.
* Step 2: $1 \times -2 = -2$. Add: $7 + (-2) = 5$.
* Step 3: $5 \times -2 = -10$. Add: $14 + (-10) = 4$.
* Step 4: $4 \times -2 = -8$. Add: $3 + (-8) = -5$.
* Result: The quotient is $x^2 + 5x + 4$ with a remainder of $-5$.
4. $(x^2 - 72) \div (x - 8)$
* Setup: Divisor $x - 8$ means use $8$. Note that the $x$ term is missing, so we must use a $0$ placeholder. Coefficients: $1, 0, -72$.
* Step 1: Bring down $1$.
* Step 2: $1 \times 8 = 8$. Add: $0 + 8 = 8$.
* Step 3: $8 \times 8 = 64$. Add: $-72 + 64 = -8$.
* Result: The quotient is $x + 8$ with a remainder of $-8$.
5. $(x^3 - 13x^2 + 49x + 15) \div (x - 7)$
* Setup: Divisor $x - 7$ means use $7$. Coefficients: $1, -13, 49, 15$.
* Step 1: Bring down $1$.
* Step 2: $1 \times 7 = 7$. Add: $-13 + 7 = -6$.
* Step 3: $-6 \times 7 = -42$. Add: $49 + (-42) = 7$.
* Step 4: $7 \times 7 = 49$. Add: $15 + 49 = 64$.
* Result: The quotient is $x^2 - 6x + 7$ with a remainder of $64$.
6. $(x^2 - 28) \div (x - 5)$
* Setup: Divisor $x - 5$ means use $5$. Missing $x$ term requires a $0$ placeholder. Coefficients: $1, 0, -28$.
* Step 1: Bring down $1$.
* Step 2: $1 \times 5 = 5$. Add: $0 + 5 = 5$.
* Step 3: $5 \times 5 = 25$. Add: $-28 + 25 = -3$.
* Result: The quotient is $x + 5$ with a remainder of $-3$.
7. $(x^3 + 30x - 18 - 4x^2) \div (x + 5)$
* Setup: First, rearrange into standard form: $x^3 - 4x^2 + 30x - 18$. Divisor $x + 5$ means use $-5$. Coefficients: $1, -4, 30, -18$.
* Step 1: Bring down $1$.
* Step 2: $1 \times -5 = -5$. Add: $-4 + (-5) = -9$.
* Step 3: $-9 \times -5 = 45$. Add: $30 + 45 = 75$.
* Step 4: $75 \times -5 = -375$. Add: $-18 + (-375) = -393$.
* Result: The quotient is $x^2 - 9x + 75$ with a remainder of $-393$.
8. $(8x + x^3 - 5x^2 + 4) \div (-1 + x)$
* Setup: Rearrange dividend: $x^3 - 5x^2 + 8x + 4$. Rearrange divisor: $x - 1$. Use $1$ on the outside. Coefficients: $1, -5, 8, 4$.
* Step 1: Bring down $1$.
* Step 2: $1 \times 1 = 1$. Add: $-5 + 1 = -4$.
* Step 3: $-4 \times 1 = -4$. Add: $8 + (-4) = 4$.
* Step 4: $4 \times 1 = 4$. Add: $4 + 4 = 8$.
* Result: The quotient is $x^2 - 4x + 4$ with a remainder of $8$.
Final Answer:
1. $x + 15, R 110$
2. $x + 5, R -10$
3. $x^2 + 5x + 4, R -5$
4. $x + 8, R -8$
5. $x^2 - 6x + 7, R 64$
6. $x + 5, R -3$
7. $x^2 - 9x + 75, R -393$
8. $x^2 - 4x + 4, R 8$
Parent Tip: Review the logic above to help your child master the concept of divide using long division worksheet.
