2.EA: Exercises for Polynomials Expressions

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Polynomials

In Exercises 1-6, state the coefficient and the degree of each of the following terms.

1) $$3 v^{5} u^{6}$$

Coefficient $$=3,$$ Degree $$=11$$

2) $$-3 b^{5} z^{8}$$

3) $$-5 v^{6}$$

Coefficient $$=-5,$$ Degree $$=6$$

4) $$-5 c^{3}$$

5) $$2 u^{7} x^{4} d^{5}$$

Coefficient $$=2,$$ Degree $$=16$$

6) $$9 w^{4} c^{5} u^{7}$$

In Exercises 7-16, state whether each of the following expressions is a monomial, binomial, or trinomial.

7) $$-7 b^{9} c^{3}$$

Monomial

8) $$7 b^{6} c^{2}$$

9) $$4 u+7 v$$

Binomial

10) $$-3 b+5 c$$

11) $$3 b^{4}-9 b c+9 c^{2}$$

Trinomial

12) $$8 u^{4}+5 u v+3 v^{4}$$

13) $$5 s^{2}+9 t^{7}$$

Binomial

14) $$-8 x^{6}-6 y^{7}$$

15) $$2 u^{3}-5 u v-4 v^{4}$$

Trinomial

16) $$6 y^{3}-4 y z+7 z^{3}$$

In Exercises 17-20, sort each of the given polynomials in descending powers of $$x$$.

17) $$-2 x^{7}-9 x^{13}-6 x^{12}-7 x^{17}$$

$$-7 x^{17}-9 x^{13}-6 x^{12}-2 x^{7}$$

18) $$2 x^{4}-8 x^{19}+3 x^{10}-4 x^{2}$$

19) $$8 x^{6}+2 x^{15}-3 x^{11}-2 x^{2}$$

$$2 x^{15}-3 x^{11}+8 x^{6}-2 x^{2}$$

20) $$2 x^{6}-6 x^{7}-7 x^{15}-9 x^{18}$$

In Exercises 21-24, sort each of the given polynomials in ascending powers of $$x$$.

21) $$7 x^{17}+3 x^{4}-2 x^{12}+8 x^{14}$$

$$3 x^{4}-2 x^{12}+8 x^{14}+7 x^{17}$$

22) $$6 x^{18}-6 x^{4}-2 x^{19}-7 x^{14}$$

23) $$2 x^{13}+3 x^{18}+8 x^{7}+5 x^{4}$$

$$5 x^{4}+8 x^{7}+2 x^{13}+3 x^{18}$$

24) $$-6 x^{18}-8 x^{11}-9 x^{15}+5 x^{12}$$

In Exercises 25-32, simplify the given polynomial, combining like terms, then arranging your answer in descending powers of $$x$$.

25) $$-5 x+3-6 x^{3}+5 x^{2}-9 x+3-3 x^{2}+6 x^{3}$$

$$2 x^{2}-14 x+6$$

26) $$-2 x^{3}+8 x-x^{2}+5+7+6 x^{2}+4 x^{3}-9 x$$

27) $$4 x^{3}+6 x^{2}-8 x+1+8 x^{3}-7 x^{2}+5 x-8$$

$$12 x^{3}-x^{2}-3 x-7$$

28) $$-8 x^{3}-2 x^{2}-7 x-3+7 x^{3}-9 x^{2}-8 x+9$$

29) $$x^{2}+9 x-3+7 x^{2}-3 x-8$$

$$8 x^{2}+6 x-11$$

30) $$-4 x^{2}-6 x+3-3 x^{2}+3 x-6$$

31) $$8 x+7+2 x^{2}-8 x-3 x^{3}-x^{2}$$

$$-3 x^{3}+x^{2}+7$$

32) $$-x^{2}+8-7 x+8 x-5 x^{2}+4 x^{3}$$

In Exercises 33-44, simplify the given polynomial, combining like terms, then arranging your answer in a reasonable order, perhaps in descending powers of either variable. Note: Answers may vary, depending on which variable you choose to dictate the order.

33) $$-8 x^{2}-4 x z-2 z^{2}-3 x^{2}-8 x z+2 z^{2}$$

$$-11 x^{2}-12 x z$$

34) $$-5 x^{2}+9 x z-4 z^{2}-6 x^{2}-7 x z+7 z^{2}$$

35) $$-6 u^{3}+4 u v^{2}-2 v^{3}-u^{3}+6 u^{2} v-5 u v^{2}$$

$$-7 u^{3}+6 u^{2} v-u v^{2}-2 v^{3}$$

36) $$7 a^{3}+6 a^{2} b-5 a b^{2}+4 a^{3}+6 a^{2} b+6 b^{3}$$

37) $$-4 b^{2} c-3 b c^{2}-5 c^{3}+9 b^{3}-3 b^{2} c+5 b c^{2}$$

$$9 b^{3}-7 b^{2} c+2 b c^{2}-5 c^{3}$$

38) $$4 b^{3}-6 b^{2} c+9 b c^{2}-9 b^{3}-8 b c^{2}+3 c^{3}$$

39) $$-8 y^{2}+6 y z-7 z^{2}-2 y^{2}-3 y z-9 z^{2}$$

$$-10 y^{2}+3 y z-16 z^{2}$$

40) $$8 x^{2}+x y+3 y^{2}-x^{2}+7 x y+y^{2}$$

41) $$7 b^{2} c+8 b c^{2}-6 c^{3}-4 b^{3}+9 b c^{2}-6 c^{3}$$

$$-4 b^{3}+7 b^{2} c+17 b c^{2}-12 c^{3}$$

42) $$7 x^{3}-9 x^{2} y+3 y^{3}+7 x^{3}+3 x y^{2}-7 y^{3}$$

43) $$9 a^{2}+a c-9 c^{2}-5 a^{2}-2 a c+2 c^{2}$$

$$4 a^{2}-a c-7 c^{2}$$

44) $$7 u^{2}+3 u v-6 v^{2}-6 u^{2}+7 u v+6 v^{2}$$

In Exercises 45-50, state the degree of the given polynomial.

45) $$3 x^{15}+4+8 x^{3}-8 x^{19}$$

$$19$$

46) $$-4 x^{6}-7 x^{16}-5+3 x^{18}$$

47) $$7 x^{10}-3 x^{18}+9 x^{4}-6$$

$$18$$

48) $$3 x^{16}-8 x^{5}+x^{8}+7$$

49) $$-2-x^{7}-5 x^{5}+x^{10}$$

$$10$$

50) $$x^{11}+7 x^{16}+8-7 x^{10}$$

51) Given $$f(x)=5 x^{3}+4 x^{2}-6$$, evaluate $$f(-1)$$.

$$-7$$

52) Given $$f(x)=-3 x^{3}+3 x^{2}-9$$, evaluate $$f(-1)$$.

53) Given $$f(x)=5 x^{4}-4 x-6$$, evaluate $$f(-2)$$.

$$82$$

54) Given $$f(x)=-2 x^{4}-4 x-9$$, evaluate $$f(2)$$.

55) Given $$f(x)=3 x^{4}+5 x^{3}-9$$, evaluate $$f(-2)$$.

$$-1$$

56) Given $$f(x)=-3 x^{4}+2 x^{3}-6$$, evaluate $$f(-1)$$.

57) Given $$f(x)=3 x^{4}-5 x^{2}+8$$, evaluate $$f(-1)$$.

$$6$$

58) Given $$f(x)=-4 x^{4}-5 x^{2}-3$$, evaluate $$f(3)$$.

59) Given $$f(x)=-2 x^{3}+4 x-9$$, evaluate $$f(2)$$.

$$-17$$

60) Given $$f(x)=4 x^{3}+3 x+7$$, evaluate $$f(-2)$$.

In Exercises 61-64, use your graphing calculator to sketch the given quadratic polynomial. In each case the graph is a parabola, so adjust the WINDOW parameters until the vertex is visible in the viewing window, as needed.

61) $$p(x)=-2 x^{2}+8 x+32$$

62) $$p(x)=2 x^{2}+6 x-18$$

63) $$p(x)=3 x^{2}-8 x-35$$

64) $$p(x)=-4 x^{2}-9 x+50$$

In Exercises 65-68, use your graphing calculator to sketch the polynomial using the given WINDOW parameters.

65) $$p(x)=x^{3}-4 x^{2}-11 x+30$$
$$\mathbf{X} \min =-10 \quad \mathbf{X} \max =10$$
$$\mathbf{Y} \min =-50 \quad \mathbf{Y} \max =50$$

66) $$p(x)=-x^{3}+4 x^{2}+27 x-90$$
$$\mathbf{X} \min =-10 \quad \mathbf{X} \max =10$$
$$\mathbf{Y} \min =-150 \quad \mathbf{Y} \max =50$$

67) $$p(x)=x^{4}-10 x^{3}-4 x^{2}+250 x-525$$
$$\mathbf{X} \min =-10 \quad \mathbf{X} \max =10$$
$$\mathbf{Y} \min =-1000 \quad \mathbf{Y} \max =500$$

68) $$p(x)=-x^{4}+2 x^{3}+35 x^{2}-36 x-180$$
$$\mathbf{X} \min =-10 \quad \mathbf{X} \max =10$$
$$\mathbf{Y} \min =-50 \quad \mathbf{Y} \max =50$$

In Exercises 1-8, simplify the given expression. Arrange your answer in some sort of reasonable order.

1) $$\left(-8 r^{2} t+7 r t^{2}+3 t^{3}\right)+\left(9 r^{3}+2 r t^{2}+4 t^{3}\right)$$

$$9 r^{3}-8 r^{2} t+9 r t^{2}+7 t^{3}$$

2) $$\left(-a^{3}-8 a c^{2}-7 c^{3}\right)+\left(-7 a^{3}-8 a^{2} c+8 a c^{2}\right)$$

3) $$\left(7 x^{2}-6 x-9\right)+\left(8 x^{2}+10 x+9\right)$$

$$15 x^{2}+4 x$$

4) $$\left(-7 x^{2}+5 x-6\right)+\left(-10 x^{2}-1\right)$$

5) $$\left(-2 r^{2}+7 r s+4 s^{2}\right)+\left(-9 r^{2}+7 r s-2 s^{2}\right)$$

$$-11 r^{2}+14 r s+2 s^{2}$$

6) $$\left(-2 r^{2}+3 r t-4 t^{2}\right)+\left(7 r^{2}+4 r t-7 t^{2}\right)$$

7) $$\left(-8 y^{3}-3 y^{2} z-6 z^{3}\right)+\left(-3 y^{3}+7 y^{2} z-9 y z^{2}\right)$$

$$-11 y^{3}+4 y^{2} z-9 y z^{2}-6 z^{3}$$

8) $$\left(7 y^{2} z+8 y z^{2}+2 z^{3}\right)+\left(8 y^{3}-8 y^{2} z+9 y z^{2}\right)$$

In Exercises 9-14, simplify the given expression by distributing the minus sign.

9) $$-\left(5 x^{2}-4\right)$$

$$-5 x^{2}+4$$

10) $$-\left(-8 x^{2}-5\right)$$

11) $$-\left(9 r^{3}-4 r^{2} t-3 r t^{2}+4 t^{3}\right)$$

$$-9 r^{3}+4 r^{2} t+3 r t^{2}-4 t^{3}$$

12) $$-\left(7 u^{3}-8 u^{2} v+6 u v^{2}+5 v^{3}\right)$$

13) $$-\left(-5 x^{2}+9 x y+6 y^{2}\right)$$

$$5 x^{2}-9 x y-6 y^{2}$$

14) $$-\left(-4 u^{2}-6 u v+5 v^{2}\right)$$

In Exercises 15-22, simplify the given expression. Arrange your answer in some sort of reasonable order.

15) $$\left(-u^{3}-4 u^{2} w+7 w^{3}\right)-\left(u^{2} w+u w^{2}+3 w^{3}\right)$$

$$-u^{3}-5 u^{2} w-u w^{2}+4 w^{3}$$

16) $$\left(-b^{2} c+8 b c^{2}+8 c^{3}\right)-\left(6 b^{3}+b^{2} c-4 b c^{2}\right)$$

17) $$\left(2 y^{3}-2 y^{2} z+3 z^{3}\right)-\left(-8 y^{3}+5 y z^{2}-3 z^{3}\right)$$

$$10 y^{3}-2 y^{2} z-5 y z^{2}+6 z^{3}$$

18) $$\left(4 a^{3}+6 a c^{2}+5 c^{3}\right)-\left(2 a^{3}+8 a^{2} c-7 a c^{2}\right)$$

19) $$\left(-7 r^{2}-9 r s-2 s^{2}\right)-\left(-8 r^{2}-7 r s+9 s^{2}\right)$$

$$r^{2}-2 r s-11 s^{2}$$

20) $$\left(-4 a^{2}+5 a b-2 b^{2}\right)-\left(-8 a^{2}+7 a b+2 b^{2}\right)$$

21) $$\left(10 x^{2}+2 x-6\right)-\left(-8 x^{2}+14 x+17\right)$$

$$18 x^{2}-12 x-23$$

22) $$\left(-5 x^{2}+19 x-5\right)-\left(-15 x^{2}+19 x+8\right)$$

In Exercises 23-28, for the given polynomial functions $$f(x)$$ and $$g(x)$$, simplify $$f(x)+g(x)$$. Arrange your answer in descending powers of $$x$$.

23) \begin{aligned}f(x)&=-2 x^{2}+9 x+7 \\ g(x)&=8 x^{3}-7 x^{2}+5\end{aligned}

$$8 x^{3}-9 x^{2}+9 x+12$$

24) \begin{aligned}f(x)&=-8 x^{3}+6 x-9 \\ g(x)&=x^{3}-x^{2}+3 x\end{aligned}

25) \begin{aligned}f(x)&=5 x^{3}-5 x^{2}+8 x \\ g(x)&=7 x^{2}-2 x-9\end{aligned}

$$5 x^{3}+2 x^{2}+6 x-9$$

26) \begin{aligned}f(x)&=-x^{2}+8 x+1 \\ g(x)&=-7 x^{3}+8 x-9\end{aligned}

27) \begin{aligned}f(x)&=-3 x^{2}-8 x-9 \\ g(x)&=5 x^{2}-4 x+4\end{aligned}

$$2 x^{2}-12 x-5$$

28) \begin{aligned}f(x)&=-3 x^{2}+x-8 \\ g(x)&=7 x^{2}-9\end{aligned}

In Exercises 29-34, for the given polynomial functions $$f(x)$$ and $$g(x)$$, simplify $$f(x)−g(x)$$. Arrange your answer in descending powers of $$x$$.

29) \begin{aligned}f(x)&=-6 x^{3}-7 x+7 \\ g(x)&=-3 x^{3}-3 x^{2}-8 x\end{aligned}

$$-3 x^{3}+3 x^{2}+x+7$$

30) \begin{aligned}f(x)&=5 x^{3}-5 x+4 \\ g(x)&=-8 x^{3}-2 x^{2}-3 x\end{aligned}

31) \begin{aligned}f(x)&=12 x^{2}-5 x+4 \\ g(x)&=8 x^{2}-16 x-7\end{aligned}

$$4 x^{2}+11 x+11$$

32) \begin{aligned}f(x)&=-7 x^{2}+12 x+17 \\ g(x)&=-10 x^{2}-17\end{aligned}

33) \begin{aligned}f(x)&=-3 x^{3}-4 x+2 \\ g(x)&=-4 x^{3}-7 x^{2}+6\end{aligned}

$$x^{3}+7 x^{2}-4 x-4$$

34) \begin{aligned}f(x)&=-9 x^{2}+9 x+3 \\ g(x)&=7 x^{3}+7 x^{2}+5\end{aligned}

In Exercises 35-36, find the area of the given square by summing the areas of its four parts.

35)

$$x^{2}+10 x+25$$

36)

Laws of Exponents

In Exercises 1-8, simplify each of the given exponential expressions.

1) $$(-4)^{3}$$

$$-64$$

2) $$(-9)^{2}$$

3) $$\left(-\dfrac{5}{7}\right)^{0}$$

$$1$$

4) $$\left(-\dfrac{2}{5}\right)^{0}$$

5) $$\left(-\dfrac{4}{3}\right)^{2}$$

$$\dfrac{16}{9}$$

6) $$\left(-\dfrac{2}{3}\right)^{2}$$

7) $$(-19)^{0}$$

$$1$$

8) $$(-17)^{0}$$

In Exercises 9-18, simplify each of the given exponential expressions.

9) $$(7 v-6 w)^{18} \cdot(7 v-6 w)^{17}$$

$$(7 v-6 w)^{35}$$

10) $$(8 a+7 c)^{3} \cdot(8 a+7 c)^{19}$$

11) $$3^{4} \cdot 3^{0}$$

$$3^{4}$$

12) $$5^{7} \cdot 5^{0}$$

13) $$4^{n} \cdot 4^{8 n+3}$$

$$4^{9 n+3}$$

14) $$4^{6 m+5} \cdot 4^{m-5}$$

15) $$x^{8} \cdot x^{3}$$

$$x^{11}$$

16) $$a^{9} \cdot a^{15}$$

17) $$2^{5} \cdot 2^{3}$$

$$2^{8}$$

18) $$2^{10} \cdot 2^{3}$$

In Exercises 19-28, simplify each of the given exponential expressions.

19) $$\dfrac{4^{16}}{4^{16}}$$

$$1$$

20) $$\dfrac{3^{12}}{3^{12}}$$

21) $$\dfrac{w^{11}}{w^{7}}$$

$$w^{4}$$

22) $$\dfrac{c^{10}}{c^{8}}$$

23) $$\dfrac{(9 a-8 c)^{15}}{(9 a-8 c)^{8}}$$

$$(9 a-8 c)^{7}$$

24) $$\dfrac{(4 b+7 c)^{15}}{(4 b+7 c)^{5}}$$

25) $$\dfrac{2^{9 n+5}}{2^{3 n-4}}$$

$$2^{6 n+9}$$

26) $$\dfrac{2^{4 k-9}}{2^{3 k-8}}$$

27) $$\dfrac{4^{17}}{4^{9}}$$

$$4^{8}$$

28) $$\dfrac{2^{17}}{2^{6}}$$

In Exercises 29-38, simplify each of the given exponential expressions.

29) $$\left(4^{8 m-6}\right)^{7}$$

$$4^{56 m-42}$$

30) $$\left(2^{2 m-9}\right)^{3}$$

31) $$\left[(9 x+5 y)^{3}\right]^{7}$$

$$(9 x+5 y)^{21}$$

32) $$\left[(4 u-v)^{8}\right]^{9}$$

33) $$\left(4^{3}\right)^{2}$$

$$4^{6}$$

34) $$\left(3^{4}\right)^{2}$$

35) $$\left(c^{4}\right)^{7}$$

$$c^{28}$$

36) $$\left(w^{9}\right)^{5}$$

37) $$\left(6^{2}\right)^{0}$$

$$1$$

38) $$\left(8^{9}\right)^{0}$$

In Exercises 39-48, simplify each of the given exponential expressions.

39) $$(u w)^{5}$$

$$u^{5} w^{5}$$

40) $$(a c)^{4}$$

41) $$(-2 y)^{3}$$

$$-8 y^{3}$$

42) $$(-2 b)^{3}$$

43) $$\left(3 w^{9}\right)^{4}$$

$$81 w^{36}$$

44) $$\left(-3 u^{9}\right)^{4}$$

45) $$\left(-3 x^{8} y^{2}\right)^{4}$$

$$81 x^{32} y^{8}$$

46) $$\left(2 x^{8} z^{6}\right)^{4}$$

47) $$\left(7 s^{6 n}\right)^{3}$$

$$343 s^{18 n}$$

48) $$\left(9 b^{6 n}\right)^{3}$$

In Exercises 49-56, simplify each of the given exponential expressions.

49) $$\left(\dfrac{v}{2}\right)^{3}$$

$$\dfrac{v^{3}}{8}$$

50) $$\left(\dfrac{t}{9}\right)^{2}$$

51) $$\left(-\dfrac{2}{u}\right)^{2}$$

$$\dfrac{4}{u^{2}}$$

52) $$\left(-\dfrac{3}{w}\right)^{3}$$

53) $$\left(-\dfrac{r^{8}}{5}\right)^{4}$$

$$\dfrac{r^{32}}{625}$$

54) $$\left(-\dfrac{x^{11}}{5}\right)^{5}$$

55) $$\left(\dfrac{5}{c^{9}}\right)^{4}$$

$$\dfrac{625}{c^{36}}$$

56) $$\left(\dfrac{5}{u^{12}}\right)^{2}$$

57) Complete each of the laws of exponents presented in the first column, then use the results to simplify the expressions in the second column.

 $$a^{m} a^{n}=?$$ $$a^{3} a^{5}=?$$ $$\dfrac{a^{m}}{a^{n}}=?$$ $$\dfrac{a^{6}}{a^{2}}=?$$ $$\left(a^{m}\right)^{n}=?$$ $$\left(a^{5}\right)^{7}=?$$ $$(a b)^{m}=?$$ $$(a b)^{9}=?$$ $$\left(\dfrac{a}{b}\right)^{m}=?$$ $$\left(\dfrac{a}{b}\right)^{3}=?$$

The general answers are: $$a^{m+n}, a^{m-n}, a^{m n}, a^{m} b^{m}, \dfrac{a^m}{b^m}$$.

The specific answers are: $$a^{8}, a^{4}, a^{35}, a^{9} b^{9}, \dfrac{a^3}{b^3}$$.

Multiplying Polynomials

In Exercises 1-10, simplify the given expression.

1) $$-3(7 r)$$

$$-21 r$$

2) $$7(3 a)$$

3) $$\left(-9 b^{3}\right)\left(-8 b^{6}\right)$$

$$72b^{9}$$

4) $$\left(8 s^{3}\right)\left(-7 s^{4}\right)$$

5) $$\left(-7 r^{2} t^{4}\right)\left(7 r^{5} t^{2}\right)$$

$$-49 r^{7} t^{6}$$

6) $$\left(-10 s^{2} t^{8}\right)\left(-7 s^{4} t^{3}\right)$$

7) $$\left(-5 b^{2} c^{9}\right)\left(-8 b^{4} c^{4}\right)$$

$$40 b^{6} c^{13}$$

8) $$\left(-9 s^{2} t^{8}\right)\left(7 s^{5} t^{4}\right)$$

9) $$\left(-8 v^{3}\right)\left(4 v^{4}\right)$$

$$-32 v^{7}$$

10) $$\left(-9 y^{3}\right)\left(3 y^{5}\right)$$

In exercises 11-50, multiply to expand the given expression.

11) $$9\left(-2 b^{2}+2 b+9\right)$$

$$-18 b^{2}+18 b+81$$

12) $$9\left(-4 b^{2}+7 b-8\right)$$

13) $$-4\left(10 t^{2}-7 t-6\right)$$

$$-40 t^{2}+28 t+24$$

14) $$-5\left(-7 u^{2}-7 u+2\right)$$

15) $$-8 u^{2}\left(-7 u^{3}-8 u^{2}-2 u+10\right)$$

$$56 u^{5}+64 u^{4}+16 u^{3}-80 u^{2}$$

16) $$-3 s^{2}\left(-7 s^{3}-9 s^{2}+6 s+3\right)$$

17) $$10 s^{2}\left(-10 s^{3}+2 s^{2}+2 s+8\right)$$

$$-100 s^{5}+20 s^{4}+20 s^{3}+80 s^{2}$$

18) $$8 u^{2}\left(9 u^{3}-5 u^{2}-2 u+5\right)$$

19) $$2 s t\left(-4 s^{2}+8 s t-10 t^{2}\right)$$

$$-8 s^{3} t+16 s^{2} t^{2}-20 s t^{3}$$

20) $$7 u v\left(-9 u^{2}-3 u v+4 v^{2}\right)$$

21) $$-2 u w\left(10 u^{2}-7 u w-2 w^{2}\right)$$

$$-20 u^{3} w+14 u^{2} w^{2}+4 u w^{3}$$

22) $$-6 v w\left(-5 v^{2}+9 v w+5 w^{2}\right)$$

23) $$(-9 x-4)(-3 x+2)$$

$$27 x^{2}-6 x-8$$

24) $$(4 x-10)(-2 x-6)$$

25) $$(3 x+8)(3 x-2)$$

$$9 x^{2}+18 x-16$$

26) $$(-6 x+8)(-x+1)$$

27) $$-12 x^{3}+14 x^{2}+6 x-5$$

$$-\dfrac{930}{289}$$

28) $$(4 x-6)\left(-7 x^{2}-10 x+10\right)$$

29) $$(x-6)\left(-2 x^{2}-4 x-4\right)$$

$$-2 x^{3}+8 x^{2}+20 x+24$$

30) $$(5 x-10)\left(-3 x^{2}+7 x-8\right)$$

31) $$(8 u-9 w)(8 u-9 w)$$

$$64 u^{2}-144 u w+81 w^{2}$$

32) $$(3 b+4 c)(-8 b+10 c)$$

33) $$(9 r-7 t)(3 r-9 t)$$

$$27 r^{2}-102 r t+63 t^{2}$$

34) $$(-6 x-3 y)(-6 x+9 y)$$

35) $$(4 r-10 s)\left(-10 r^{2}+10 r s-7 s^{2}\right)$$

$$-40 r^{3}+140 r^{2} s-128 r s^{2}+70 s^{3}$$

36) $$(5 s-9 t)\left(-3 s^{2}+4 s t-9 t^{2}\right)$$

37) $$(9 x-2 z)\left(4 x^{2}-4 x z-10 z^{2}\right)$$

$$36 x^{3}-44 x^{2} z-82 x z^{2}+20 z^{3}$$

38) $$(r-4 t)\left(7 r^{2}+4 r t-2 t^{2}\right)$$

39) $$(9 r+3 t)^{2}$$

$$81 r^{2}+54 r t+9 t^{2}$$

40) $$(4 x+8 z)^{2}$$

41) $$(4 y+5 z)(4 y-5 z)$$

$$16 y^{2}-25 z^{2}$$

42) $$(7 v+2 w)(7 v-2 w)$$

43) $$(7 u+8 v)(7 u-8 v)$$

$$49 u^{2}-64 v^{2}$$

44) $$(6 b+8 c)(6 b-8 c)$$

45) $$(7 b+8 c)^{2}$$

$$49 b^{2}+112 b c+64 c^{2}$$

46) $$(2 b+9 c)^{2}$$

47) $$\left(2 t^{2}+9 t+4\right)\left(2 t^{2}+9 t+4\right)$$

$$4 t^{4}+36 t^{3}+97 t^{2}+72 t+16$$

48) $$\left(3 a^{2}-9 a+4\right)\left(3 a^{2}-9 a+2\right)$$

49) $$\left(4 w^{2}+3 w+5\right)\left(3 w^{2}-6 w+8\right)$$

$$12 w^{4}-15 w^{3}+29 w^{2}-6 w+40$$
50) $$\left(4 s^{2}+3 s+8\right)\left(2 s^{2}+4 s-9\right)$$