Solutions to Problem Set 11

Math 101-06

10-2-2017

[Polynomials]

1. Compute $(2 x^3 + 5 x^2 - 7 x
   + 3) + (6 x^3 - 8 x + 10)$ .

$$(2 x^3 + 5 x^2 - 7 x + 3) + (6 x^3 - 8 x + 10) = 8 x^3 + 5 x^2 - 15 x + 13.\quad\halmos$$


2. Compute $(7 x^3 - 5 x - 8) -
   (2 x^3 + x^2 + x - 10)$ .

$$(7 x^3 - 5 x - 8) - (2 x^3 + x^2 + x - 10) = 5 x^3 - x^2 - 6 x + 2.\quad\halmos$$


3. Compute $6(2 x + 1) + 3(x^2 +
   x + 5)$ .

$$6(2 x + 1) + 3(x^2 + x + 5) = (12 x + 6) + (3 x^2 + 3 x + 15) = 3 x^2 + 15 x + 21.\quad\halmos$$


4. Compute $(x - 8)(x + 3)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & x & & 3 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & x & & $x^2$ & & $3 x$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & -8 & & $-8 x$ & & -24 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(x - 8)(x + 3) = x^2 - 5 x - 24.\quad\halmos$$


5. Compute $(2 x + 1)(3 x + 5)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & $2 x$ & & 1 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $3 x$ & & $6 x^2$ & & $3 x$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 5 & & $10 x$ & & 5 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(2 x + 1)(x + 5) = 6 x^2 + 13 x + 5.\quad\halmos$$


6. Compute $(3 x^2 + x - 1)(2 x
   + 7)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & \cr & & & $3 x^2$ & & x & & -1 & \cr height2pt & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & \cr & $2 x$ & & $6 x^3$ & & $2 x^2$ & & $-2 x$ & \cr height2pt & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & & \omit & \cr & 7 & & $21 x^2$ & & $7 x$ & & -7 & \cr height2pt & \omit & & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(3 x^2 + x - 1)(2 x + 7) = 6 x^3 + 23 x^2 + 5 x - 7.\quad\halmos$$


7. Compute $(2 x + y)(x^2 - x
   y)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & $2 x$ & & y & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $x^2$ & & $2 x^3$ & & $x^2 y$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $-x y$ & & $-2 x^2 y$ & & $-x y^2$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}$$

$$(2 x + y)(x^2 - x y) = 2 x^3 - x^2 y - x y^2.\quad\halmos$$


8. Compute $(4 x^2 + 3)(2 x -
   7)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & $2 x$ & & -7 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $4 x^2$ & & $8 x^3$ & & $-28 x^2$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 3 & & $6 x$ & & -21 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(4 x^2 + 3)(2 x - 7) = 8 x^3 - 28 x^2 + 6 x - 21.\quad\halmos$$


9. Compute $(x + 5)(x - 5)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & x & & -5 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & x & & $x^2$ & & $-5 x$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 5 & & $5 x$ & & -25 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(x + 5)(x - 5) = x^2 - 25.\quad\halmos$$


10. Compute $(x + 5)^2$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & x & & 5 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & x & & $x^2$ & & $5 x$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 5 & & $5 x$ & & 25 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(x + 5)^2 = x^2 + 10 x + 25.\quad\halmos$$


11. Compute $(x - 6)^2$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & x & & -6 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & x & & $x^2$ & & $-6 x$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & -6 & & $-6 x$ & & 36 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(x - 6)^2 = x^2 - 12 x + 36.\quad\halmos$$


12. Compute $(2 x - 3)(2 x +
   3)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & $2 x$ & & 3 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $2 x$ & & $4 x^2$ & & $6 x$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & -3 & & $-6 x$ & & -9 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(2 x - 3)(2 x + 3) = 4 x^2 - 9.\quad\halmos$$


13. Compute $(5 x + 2 y)(5 x - 2
   y)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & $5 x$ & & $2 y$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $5 x$ & & $25 x^2$ & & $10 x y$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $-2 y$ & & $-10 x y$ & & $-4 y^2$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(5 x + 2 y)(5 x - 2 y) = 25 x^2 - 4 y^2.\quad\halmos$$


14. Compute $(x^2 + 3 x + 5)(2 x
   + 3)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & $2 x$ & & 3 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $x^2$ & & $2 x^3$ & & $3 x^2$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $3 x$ & & $6 x^2$ & & $9 x$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 5 & & $10 x$ & & 15 & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(x^2 + 3 x + 5)(2 x + 3) = 2 x^3 + 9 x^2 + 19 x + 15.\quad\halmos$$


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