Dividing Polynomials: 12x^3+16x^2+7x+1 divided by 3x+1
In this article, we will explore the process of dividing polynomials, specifically dividing the polynomial 12x^3+16x^2+7x+1 by 3x+1.
What is Polynomial Division?
Polynomial division is a process of dividing one polynomial by another, resulting in a quotient and a remainder. It is similar to dividing integers, but with polynomials, we need to follow specific rules to ensure that the division is done correctly.
Step-by-Step Division Process
To divide the polynomial 12x^3+16x^2+7x+1 by 3x+1, we will follow these steps:
Step 1: Write the dividend and divisor
The dividend is the polynomial being divided, which is 12x^3+16x^2+7x+1. The divisor is the polynomial by which we are dividing, which is 3x+1.
Step 2: Divide the leading terms
Divide the leading term of the dividend (12x^3) by the leading term of the divisor (3x). This gives us 4x^2.
Step 3: Multiply the result by the divisor
Multiply the result from step 2 (4x^2) by the divisor (3x+1). This gives us 12x^3 + 4x^2.
Step 4: Subtract the product from the dividend
Subtract the product from step 3 from the dividend. This gives us 12x^2 + 7x + 1.
Step 5: Repeat steps 2-4
Repeat steps 2-4 with the new polynomial from step 4.
Divide the leading term of the new polynomial (12x^2) by the leading term of the divisor (3x). This gives us 4x.
Multiply the result by the divisor. This gives us 12x^2 + 4x.
Subtract the product from the new polynomial. This gives us 3x + 1.
Step 6: Write the final result
The final result is the quotient and remainder. The quotient is the result of the repeated division process, which is 4x^2 - 4x + 1. The remainder is the final result from step 5, which is 3x + 1.
Final Result
The final result of dividing 12x^3+16x^2+7x+1 by 3x+1 is:
Quotient: 4x^2 - 4x + 1 Remainder: 3x + 1
Therefore, we can write the division as:
12x^3+16x^2+7x+1 = (3x+1)(4x^2 - 4x + 1) + (3x + 1)
