Use long division to divide the polynomial 𝑓(𝑥) by the polynomial 𝑔(𝑥). Then write your work in the form "dividend = quotient × divisor + remainder"

Question

𝑓(𝑥)=2𝑥^4+3𝑥^2−4𝑥−5, 𝑔(𝑥)=𝑥^2+3

Answer 1

(2𝑥^4 + 3𝑥^2 − 4𝑥 − 5) ÷ (𝑥^2 + 3)

Look at the leading term for both the dividend and the divisor. The term containing the highest power of the variable is called the leading term.

The leading term for dividend is 2𝑥^4

The leading term for divisor is 𝑥^2

Step 1) See what term multiplies by x^2 to get to 2x^4. That would be 2x^2.

Step 2) Multiply 2x^2 by (x^2 + 3), that would be 2x^4 + 6x^2

Step 3) Subtract (2x^4 + 6x^2) from (2𝑥^4 + 3𝑥^2 − 4𝑥 − 5)

(2𝑥^4 + 3𝑥^2 − 4𝑥 − 5) - (2x^4 + 6x^2)

That gives us -3x^2 - 4x - 5. The leading term for the dividend now is -3x^2

Step 4) See what term multiplies by x^2 to get to -3x^2. That would be -3.

Step 5) Multiply -3 by (x^2 + 3), that would be -3x^2 - 9

Step 6) Subtract (-3x^2 - 9) from (-3x^2 - 4x - 5)

(-3x^2 - 4x - 5) - (-3x^2 - 9)

That gives us -4x + 4.

Since the leading term for the divisor x^2 has a higher power than the leading term for the dividend -4x, we stop doing long division and consider -4x + 4 to be our remainder.

Take our bold numbers: 2x^2, - 3 and place them as our quotient 2x^2 - 3.

We express (2𝑥^4 + 3𝑥^2 − 4𝑥 − 5) = (2x^2 - 3)(x^2 + 3) + (-4x + 4) as our final solution

Bonus: To do a check, expand the right side of the equation: (2x^2 - 3)(x^2 + 3) + (-4x + 4) to make sure it equals the left side

Using FOIL (First, Outer, Inner, Last) to expand (2x^2 - 3)(x^2 + 3),

(2x^2 - 3)(x^2 + 3) is the same as:

F: 2x^2 · x^2 = 2x^4

O: 2x^2 · 3 = 6x^2

I: -3 · x^2 = -3x^2

L: -3 · 3 = -9

Combine the bolded values we have 2x^4 + 6x^2 - 3x^2 - 9.

Don't forget to add (-4x + 4)

2x^4 + 6x^2 - 3x^2 - 9 + (-4x + 4)

same as 2x^4 + 6x^2 - 3x^2 - 9 - 4x + 4

Combine like terms: 6x^2 - 3x^2 and -9 + 4

We get 2x^4 + 3x^2 - 4x - 5, the same expression as the left side. Hope this helps!