To find the remainder when the polynomial x³ + 4x² + 12x + 9 is divided by x², we can use polynomial long division or simply substitute the variable to evaluate the polynomial’s remainder directly.
Since we’re dividing by x², the remainder will be a polynomial of degree less than 2. This means we can express the remainder in the form:
R(x) = ax + bwhere a and b are constants. Now, we can express the original polynomial as:
x³ + 4x² + 12x + 9 = Q(x) * x² + R(x)where Q(x) is the quotient polynomial.
Next, we carry out polynomial division:
- 1. Divide the leading term of the dividend, x³, by the leading term of the divisor, x², which gives us x.
- 2. Multiply x by the entire divisor x² to get x³ and subtract it from the original polynomial:
(x³ + 4x² + 12x + 9) - (x³) = 4x² + 12x + 9- 3. Now, take the resulting polynomial 4x² + 12x + 9 and divide the leading term 4x² by x², which results in 4.
- 4. Multiply 4 by x² and subtract:
(4x² + 12x + 9) - (4x²) = 12x + 95. Now, the next step is to realize that the remainder will no longer have terms that can be divided by x². This leads us to the conclusion that:
R(x) = 12x + 9Thus, the remainder when x³ + 4x² + 12x + 9 is divided by x² is:
12x + 9In summary, the final answer is:
12x + 9This remainder gives us the leftover portion of the original polynomial when divided by x².