1. (3x³ + 4x² - 5x - 30) ÷ (x - 2) =
2. (-4x³ + 6x² + 4x - 6) ÷ (2x - 3) =
3. (2x³ - 3x² + 17x + 39) ÷ (2x + 3) =
4. (x³ + x² + 3x - 18) ÷ (x - 2) =
5. (x³ - 2x² + 16) ÷ (x + 2) =
6. (4x³ - 2x² + 2x - 1) ÷ (2x - 1) =
7. (5x³ - 2x² + 5x - 2) ÷ (5x - 2) =
8. (x³ - 2x² - 2x + 12) ÷ (x + 2) =
Answer (2)
Let's break down each polynomial division step-by-step using synthetic division, as it's a suitable method for these types of problems.
( 3 x 3 + 4 x 2 − 5 x − 30 ) ÷ ( x − 2 ) :
Step 1: Write down the coefficients of the dividend: 3 , 4 , − 5 , − 30 .
Step 2: The divisor x − 2 has a root at 2 .
Step 3: Bring down the leading coefficient 3 .
Step 4: Multiply 3 by 2 and add to the next coefficient 4 . This gives you 10 .
Step 5: Multiply 10 by 2 and add to the next coefficient − 5 . This gives you 15 .
Step 6: Multiply 15 by 2 and add to the next coefficient − 30 . This gives you 0 .
The quotient is 3 x 2 + 10 x + 15 with a remainder of 0 .
( − 4 x 3 + 6 x 2 + 4 x − 6 ) ÷ ( 2 x − 3 ) :
Step 1: Write down the coefficients: − 4 , 6 , 4 , − 6 .
Step 2: The divisor 2 x − 3 has a root at 2 3 .
Step 3: Bring down the leading coefficient − 4 .
Step 4: Multiply − 4 by 2 3 and add to the next coefficient 6 . This gives you 0 .
Step 5: Repeat the process for the remaining coefficients.
Calculate these steps to find the quotient and remainder.
( 2 x 3 − 3 x 2 + 17 x + 39 ) ÷ ( 2 x + 3 ) :
Step 1: Write down the coefficients: 2 , − 3 , 17 , 39 .
Step 2: The divisor 2 x + 3 has a root at − 2 3 .
Step 3: Bring down the leading coefficient 2 .
Step 4: Multiply 2 by − 2 3 and add to the next coefficient.
Proceed with these calculations to find the quotient and remainder.
( x 3 + x 2 + 3 x − 18 ) ÷ ( x − 2 ) :
Step 1: The divisor x − 2 has a root at 2 .
Step 2: Bring down the leading coefficient 1 and follow the synthetic division process.
Complete the operations to discover the quotient and remainder.
Continue similarly for the remaining problems by identifying the root of the divisor, performing synthetic division, and calculating the results. Remember that the divisor will give the constant added or subtracted to the outcomes at each step.
Synthetic division provides a streamlined way to divide polynomials by linear binomials and is particularly useful in simplifying quadratic or cubic expressions when one knows at least one root.
To solve the polynomial divisions using synthetic division, you start by identifying the coefficients and the root of the divisor. Then, systematically bring down coefficients, multiply, and combine them to get the quotient and remainder. This method efficiently simplifies polynomial division.
;
