2.EA: Exercises for Polynomials Expressions

Polynomials

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

1) \(3 v^{5} u^{6}\)

Answer

Coefficient \(=3,\) Degree \(=11\)

2) \(-3 b^{5} z^{8}\)

3) \(-5 v^{6}\)

Answer

Coefficient \(=-5,\) Degree \(=6\)

4) \(-5 c^{3}\)

5) \(2 u^{7} x^{4} d^{5}\)

Answer

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) \(-7 b^{9} c^{3}\)

Answer

Monomial

8) \(7 b^{6} c^{2}\)

9) \(4 u+7 v\)

Answer

Binomial

10) \(-3 b+5 c\)

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

Answer

Trinomial

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

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

Answer

Binomial

14) \(-8 x^{6}-6 y^{7}\)

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

Answer

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

Answer

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

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

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

Answer

\(2 x^{15}-3 x^{11}+8 x^{6}-2 x^{2}\)

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) \(7 x^{17}+3 x^{4}-2 x^{12}+8 x^{14}\)

Answer

\(3 x^{4}-2 x^{12}+8 x^{14}+7 x^{17}\)

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

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

Answer

\(5 x^{4}+8 x^{7}+2 x^{13}+3 x^{18}\)

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) \(-5 x+3-6 x^{3}+5 x^{2}-9 x+3-3 x^{2}+6 x^{3}\)

Answer

\(2 x^{2}-14 x+6\)

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

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

Answer

\(12 x^{3}-x^{2}-3 x-7\)

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

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

Answer

\(8 x^{2}+6 x-11\)

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

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

Answer

\(-3 x^{3}+x^{2}+7\)

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) \(-8 x^{2}-4 x z-2 z^{2}-3 x^{2}-8 x z+2 z^{2}\)

Answer

\(-11 x^{2}-12 x z\)

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

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

Answer

\(-7 u^{3}+6 u^{2} v-u v^{2}-2 v^{3}\)

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

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

Answer

\(9 b^{3}-7 b^{2} c+2 b c^{2}-5 c^{3}\)

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

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

Answer

\(-10 y^{2}+3 y z-16 z^{2}\)

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

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

Answer

\(-4 b^{3}+7 b^{2} c+17 b c^{2}-12 c^{3}\)

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

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

Answer

\(4 a^{2}-a c-7 c^{2}\)

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) \(3 x^{15}+4+8 x^{3}-8 x^{19}\)

Answer

\(19\)

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

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

Answer

\(18\)

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

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

Answer

\(10\)

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

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

Answer

\(-7\)

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 \(f(-2)\).

Answer

\(82\)

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 \(f(-2)\).

Answer

\(-1\)

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 \(f(-1)\).

Answer

\(6\)

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 \(f(2)\).

Answer

\(-17\)

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

Answer

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

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

Answer

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

Answer

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

Answer

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

Adding and Subtracting Polynomials

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

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

Answer

\(9 r^{3}-8 r^{2} t+9 r t^{2}+7 t^{3}\)

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

Answer

\(15 x^{2}+4 x\)

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

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

Answer

\(-11 r^{2}+14 r s+2 s^{2}\)

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

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

Answer

\(-11 y^{3}+4 y^{2} z-9 y z^{2}-6 z^{3}\)

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

Answer

\(-5 x^{2}+4\)

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

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

Answer

\(-9 r^{3}+4 r^{2} t+3 r t^{2}-4 t^{3}\)

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

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

Answer

\(5 x^{2}-9 x y-6 y^{2}\)

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

Answer

\(-u^{3}-5 u^{2} w-u w^{2}+4 w^{3}\)

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

Answer

\(10 y^{3}-2 y^{2} z-5 y z^{2}+6 z^{3}\)

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

Answer

\(r^{2}-2 r s-11 s^{2}\)

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

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

Answer

\(18 x^{2}-12 x-23\)

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

Answer

\(8 x^{3}-9 x^{2}+9 x+12\)

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

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

Answer

\(5 x^{3}+2 x^{2}+6 x-9\)

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

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

Answer

\(2 x^{2}-12 x-5\)

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

Answer

\(-3 x^{3}+3 x^{2}+x+7\)

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

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

Answer

\(4 x^{2}+11 x+11\)

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

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

Answer

\(x^{3}+7 x^{2}-4 x-4\)

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)

Answer

\(x^{2}+10 x+25\)

36)

Laws of Exponents

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

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

Answer

\(-64\)

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

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

Answer

\(1\)

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

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

Answer

\(\dfrac{16}{9}\)

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

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

Answer

\(1\)

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

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

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

Answer

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

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

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

Answer

\(3^{4}\)

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

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

Answer

\(4^{9 n+3}\)

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

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

Answer

\(x^{11}\)

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

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

Answer

\(2^{8}\)

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

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

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

Answer

\(1\)

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

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

Answer

\(w^{4}\)

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

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

Answer

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

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

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

Answer

\(2^{6 n+9}\)

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

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

Answer

\(4^{8}\)

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

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

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

Answer

\(4^{56 m-42}\)

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

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

Answer

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

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

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

Answer

\(4^{6}\)

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

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

Answer

\(c^{28}\)

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

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

Answer

\(1\)

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

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

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

Answer

\(u^{5} w^{5}\)

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

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

Answer

\(-8 y^{3}\)

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

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

Answer

\(81 w^{36}\)

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

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

Answer

\(81 x^{32} y^{8}\)

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

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

Answer

\(343 s^{18 n}\)

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

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

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

Answer

\(\dfrac{v^{3}}{8}\)

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

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

Answer

\(\dfrac{4}{u^{2}}\)

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

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

Answer

\(\dfrac{r^{32}}{625}\)

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

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

Answer

\(\dfrac{625}{c^{36}}\)

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.

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

Answer

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) \(-3(7 r)\)

Answer

\(-21 r\)

2) \(7(3 a)\)

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

Answer

\(72b^{9}\)

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

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

Answer

\(-49 r^{7} t^{6}\)

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

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

Answer

\(40 b^{6} c^{13}\)

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

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

Answer

\(-32 v^{7}\)

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

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

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

Answer

\(-18 b^{2}+18 b+81\)

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

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

Answer

\(-40 t^{2}+28 t+24\)

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

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

Answer

\(56 u^{5}+64 u^{4}+16 u^{3}-80 u^{2}\)

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

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

Answer

\(-100 s^{5}+20 s^{4}+20 s^{3}+80 s^{2}\)

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

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

Answer

\(-8 s^{3} t+16 s^{2} t^{2}-20 s t^{3}\)

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

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

Answer

\(-20 u^{3} w+14 u^{2} w^{2}+4 u w^{3}\)

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

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

Answer

\(27 x^{2}-6 x-8\)

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

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

Answer

\(9 x^{2}+18 x-16\)

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

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

Answer

\(-\dfrac{930}{289}\)

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

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

Answer

\(-2 x^{3}+8 x^{2}+20 x+24\)

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

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

Answer

\(64 u^{2}-144 u w+81 w^{2}\)

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

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

Answer

\(27 r^{2}-102 r t+63 t^{2}\)

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

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

Answer

\(-40 r^{3}+140 r^{2} s-128 r s^{2}+70 s^{3}\)

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

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

Answer

\(36 x^{3}-44 x^{2} z-82 x z^{2}+20 z^{3}\)

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

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

Answer

\(81 r^{2}+54 r t+9 t^{2}\)

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

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

Answer

\(16 y^{2}-25 z^{2}\)

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

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

Answer

\(49 u^{2}-64 v^{2}\)

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

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

Answer

\(49 b^{2}+112 b c+64 c^{2}\)

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

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

Answer

\(4 t^{4}+36 t^{3}+97 t^{2}+72 t+16\)

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

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

Answer

\(12 w^{4}-15 w^{3}+29 w^{2}-6 w+40\)

50) \(\left(4 s^{2}+3 s+8\right)\left(2 s^{2}+4 s-9\right)\)