Algebra · real student question

Perform the indicated operation and state the degree: (4x^3 - 3x^2 + 2x - 8) - (7x^3 - 9x^2 - 5x + 10).

Question

Perform the indicated operation and indicate the resulting polynomial's degree, writing the answer in standard form: (4x3−3x2+2x−8)−(7x3−9x2−5x+10).\left(4x^{3}-3x^{2}+2x-8\right)-\left(7x^{3}-9x^{2}-5x+10\right).

Step-by-step solution

  1. Turn the subtraction into an addition. Subtracting a polynomial means adding the opposite of every one of its terms, so all four signs in the second bracket flip: 4x3−3x2+2x−8−7x3+9x2+5x−10.4x^{3}-3x^{2}+2x-8-7x^{3}+9x^{2}+5x-10 . Flipping only the first sign is the standard mistake here.

  2. Combine the cubic and quadratic terms. 4x3−7x3=−3x3,−3x2+9x2=6x2.4x^{3}-7x^{3}=-3x^{3},\qquad -3x^{2}+9x^{2}=6x^{2}.

  3. Combine the linear and constant terms. 2x+5x=7x,−8−10=−18.2x+5x=7x,\qquad -8-10=-18 .

  4. Write the result in standard form. Ordering by descending powers gives −3x3+6x2+7x−18.-3x^{3}+6x^{2}+7x-18 .

  5. Read off the degree. The highest power present is x3x^{3} with a nonzero coefficient −3-3, so the degree is 33. Note the cubic terms did not cancel, which is why the degree did not drop below 33.

Answer

−3x3+6x2+7x−18,degree 3-3x^{3}+6x^{2}+7x-18,\qquad \text{degree } 3

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