Algebra · real student question

Subtract the polynomials (x^3 - 7xy - 9y^2) - (2x^3 + 3xy - 7y^2) and state the degree of the resulting polynomial.

Question

Subtract the polynomials and indicate the degree of the resulting polynomial:

(x3−7xy−9y2)−(2x3+3xy−7y2)\left(x^3-7xy-9y^2\right)-\left(2x^3+3xy-7y^2\right)

Step-by-step solution

  1. Change the subtraction into an addition. Subtracting a polynomial means adding its opposite, so every sign inside the second bracket flips:

    −(2x3+3xy−7y2)=−2x3−3xy+7y2.-\left(2x^{3}+3xy-7y^{2}\right)=-2x^{3}-3xy+7y^{2}.

    The −7y2-7y^2 becoming +7y2+7y^2 is the term most often left unchanged by mistake.

  2. Write the expression with no brackets.

    x3−7xy−9y2−2x3−3xy+7y2.x^{3}-7xy-9y^{2}-2x^{3}-3xy+7y^{2}.

  3. Group like terms. In two variables, like terms must match in both exponents:

    (x3−2x3)+(−7xy−3xy)+(−9y2+7y2).\left(x^{3}-2x^{3}\right)+\left(-7xy-3xy\right)+\left(-9y^{2}+7y^{2}\right).

  4. Combine each group.

    −x3−10xy−2y2.-x^{3}-10xy-2y^{2}.

  5. Read off the degree. The degree of a multivariable term is the sum of its exponents: x3x^3 has degree 33, xyxy has degree 1+1=21+1=2, and y2y^2 has degree 22. The largest is 33, and its term did not cancel, so the difference is a degree 3 polynomial.

  6. Spot-check with numbers. At x=1,y=1x=1,y=1 the original is (1−7−9)−(2+3−7)=−15−(−2)=−13(1-7-9)-(2+3-7)=-15-(-2)=-13, and the answer gives −1−10−2=−13-1-10-2=-13. At x=2,y=−1x=2,y=-1 the original is (8+14−9)−(16−6−7)=13−3=10(8+14-9)-(16-6-7)=13-3=10, and the answer gives −8+20−2=10-8+20-2=10.

Answer

−x3−10xy−2y2,degree 3-x^{3}-10xy-2y^{2},\qquad \text{degree }3

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