Question

(3x4+9x2−13)÷(x2−5x). Complete the division process and find the remainder of this polynomial division.

Answer 1

To find the remainder of this polynomial division, we need to divide the given polynomial, which is 3x^4 + 9x^2 - 13, by the divisor, which is x^2 - 5x.

The first step is to rewrite the polynomial division problem in the correct form:

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x^2 - 5x | 3x^4 + 9x^2 + 0x - 13

To start dividing, we will compare the first term of the dividend (3x^4) with the first term of the divisor (x^2).

The quotient will be (3x^4/x^2), which is 3x^2. We then multiply this quotient by the divisor, which gives us the product (3x^2 * (x^2 - 5x)), which is 3x^4 - 15x^3.

Next, we subtract this product from the dividend:

3x^4 + 9x^2 - 13 - (3x^4 - 15x^3)

Simplifying, we get:

3x^4 + 9x^2 - 13 - 3x^4 + 15x^3

Combining like terms, we have:

15x^3 + 9x^2 - 13

We now repeat the process, comparing the first term of the new dividend (15x^3) with the first term of the divisor (x^2).

The new quotient will be (15x^3/x^2), which is 15x. We then multiply this quotient by the divisor, which gives us the product (15x * (x^2 - 5x)), which is 15x^3 - 75x^2.

Subtracting this product from the new dividend, we have:

15x^3 + 9x^2 - 13 - (15x^3 - 75x^2)

Simplifying, we get:

15x^3 + 9x^2 - 13 - 15x^3 + 75x^2

Combining like terms once again, we have:

84x^2 - 13.

Since there are no more terms with a degree higher than the divisor, we stop the division process.

Therefore, the remainder of this polynomial division is 84x^2 - 13.