4.10: Exercise Supplement

Algebraic Expressions

For the following problems, write the number of terms that appear, then write the terms.

Exercise $\PageIndex{1}$

$4x^2 + 7x + 12$

Answer

three: $4x^2, 7x, 12$

Exercise $\PageIndex{2}$

$14y^6$

Exercise $\PageIndex{3}$

$c + 8$

Answer

two: $c, 8$

Exercise $\PageIndex{4}$

$8$

List, if any should appear, the common factors for the following problems.

Exercise $\PageIndex{5}$

$a^2 + 4a^2 + 6a^2$

Answer

$a^2$

Exercise $\PageIndex{6}$

$9y^4 - 18y^4$

Exercise $\PageIndex{7}$

$12x^2y^3 + 36y^3$

Answer

$12y^3$

Exercise $\PageIndex{8}$

$6(a+4) + 12(a+4)$

Exercise $\PageIndex{9}$

$4(a+2b)+6(a+2b)$

Answer

$2(a+2b)$

Exercise $\PageIndex{10}$

$17x^2y(z+4) + 51y(z+4)$

Exercise $\PageIndex{11}$

$6a^2b^3c + 5x^2y$

Answer

no common factors

For the following problems, answer the question of how many.

Exercise $\PageIndex{12}$

$x$'s in $9x$?

Exercise $\PageIndex{13}$

$(a+b)$'s in $12(a+b)$?

Answer

12

Exercise $\PageIndex{14}$

$a^4$'s in $6a^4$

Exercise $\PageIndex{15}$

$c^3$'s in $2a^2bc^3$?

Answer

$2a^2b$

Exercise $\PageIndex{16}$

$(2x+3y)^2$'s in $5(x+2y)(2x+3y)^3$?

For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors.

Exercise $\PageIndex{17}$

$8z, z$

Answer

$8$

Exercise $\PageIndex{18}$

$16a^3b^2c^4, c^4$

Exercise $\PageIndex{19}$

$7y(y+3), 7y$

Answer

$(y+3)$

Exercise $\PageIndex{20}$

$(-5)a^5b^5c^5, bc$

Equations

For the following problems, observe the equations and write the relationship being expressed.

Exercise $\PageIndex{21}$

$a = 3b$

Answer

The value of $a$ is equal to three times the value of $b$.

Exercise $\PageIndex{22}$

$r = 4t + 11$

Exercise $\PageIndex{23}$

$f = \dfrac{1}{2}m^2 + 6g$

Answer

The value of $f$ is equal to six times $g$ more then one half times the value of $m$ squared.

Exercise $\PageIndex{24}$

$x = 5y^3 + 2y + 6$

Exercise $\PageIndex{25}$

$P^2 = ka^3$

Answer

The value of $P$ squared is equal to the value of $a$ cubed times $k$.

Use numerical evaluation to evaluate the equations for the following problems.

Exercise $\PageIndex{26}$

$C = 2 \pi r$. Find $C$ is $\pi$ is approximated by $3.14$ and $r = 6$

Exercise $\PageIndex{27}$

$I = \dfrac{E}{R}$. Find $I$ is $E = 20$ and $R = 2$.

Answer

$10$

Exercise $\PageIndex{28}$

$I=prt$. Find $I$ if $p=1000$, $r=0.06$, and $t=3$.

Exercise $\PageIndex{29}$

$E = mc^2$. Find $E$ if $m = 120$ and $c = 186,000$.

Answer

$4.1515 \times 10^{12}$

Exercise $\PageIndex{30}$

$z = \dfrac{x-u}{s}$. Find $z$ if $x = 42$, $u = 30$, and $s = 12$.

Exercise $\PageIndex{31}$

$R = \dfrac{24C}{P(n+1)}$. Find $R$ if $C = 35$, $P = 300$, and $n = 19$.

Answer

$\dfrac{7}{50}$ or $0.14$

Classification of Expressions and Equations

For the following problems, classify each of the polynomials as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term.

Exercise $\PageIndex{32}$

$2a+9$

Exercise $\PageIndex{33}$

$4y^3 + 3y + 1$

Answer

trinomial, cubic; 4, 3, 1

Exercise $\PageIndex{34}$

$10a^4$

Exercise $\PageIndex{35}$

$147$

Answer

monomial; zero; 147

Exercise $\PageIndex{36}$

$4xy + 2yz^2 + 6x$

Exercise $\PageIndex{37}$

$9ab^2c^2 + 10a^3b^2c^5$

Answer

binomial; tenth; 9, 10

Exercise $\PageIndex{38}$

$(2xy^3)^0, xy^3 \not = 0$

Exercise $\PageIndex{39}$

Why is the expression $\dfrac{4x}{3x-7}$ not a polynomial?

Answer

... because there is a variable in the denominator

Exercise $\PageIndex{40}$

Why is the expression $5a^{\dfrac{3}{4}}$ not a polynomial?

For the following problems, classify each of the equations by degree. If the term linear, quadratic, or cubic applies, use it.

Exercise $\PageIndex{41}$

$3y + 2x = 1$

Answer

linear

Exercise $\PageIndex{42}$

$4a^2 - 5a + 8 = 0$

Exercise $\PageIndex{43}$

$y - x - z + 4w = 21$

Answer

linear

Exercise $\PageIndex{44}$

$5x^2 + 2x^2 - 3x + 1 = 19$

Exercise $\PageIndex{45}$

$(6x^3)^0 + 5x^2 = 7$

Answer

Quadratic

Combining Polynomials Using Addition and Subtraction- Special Binomial Products

Simplify the algebraic expressions for the following problems.

Exercise $\PageIndex{46}$

$4a^2b + 8a^2b - a^2b$

Exercise $\PageIndex{47}$

$21x^2y^3 + 3xy + x^2y^3 + 6$

Answer

$22x^2y^3 + 3xy + 6$

Exercise $\PageIndex{48}$

$7(x+1)+2x−6$

Exercise $\PageIndex{49}$

$2(3y^2+4y+4)+5y^2+3(10y+2)$

Answer

$11y^2 + 38y + 14$

Exercise $\PageIndex{50}$

$5[3x + 7(2x^2 + 3x + 2) + 5] - 10x^2 + 4(3x^2 + x)$

Exercise $\PageIndex{51}$

$8{3[4y^3+y+2] + 6(y^3+2y^2)} - 24y^3 - 10y^2 - 3$

Answer

$120y^3 + 86y^2 + 24y + 45$

Exercise $\PageIndex{52}$

$4a^2bc^3 + 5abc^3 + 9abc^3 + 7a^2bc^2$

Exercise $\PageIndex{53}$

$x(2x+5) + 3x^2 - 3x + 3$

Answer

$5x^2 + 2x + 3$

Exercise $\PageIndex{54}$

$4k(3k^2 + 2k + 6) + k(5k^2 + k) + 16$

Exercise $\PageIndex{55}$

$2{5[6(b+2a+c^2)]}$

Answer

$60c^2 + 120a + 60b$

Exercise $\PageIndex{56}$

$9x^2y(3xy + 4x) - 7x^3y^2 - 30x^3y + 5y(x^3y + 2x)$

Exercise $\PageIndex{57}$

$3m[5 + 2m(m+6m^2)] + m(m^2 + 4m + 1)$

Answer

$36m^4 + 7m^3 + 4m^2 + 16m$

Exercise $\PageIndex{58}$

$2r[4(r + 5) - 2r - 10] + 6r(r + 2)$

Exercise $\PageIndex{59}$

$abc(3abc + c + b) + 6a(2bc + bc^2)$

Answer

$3a^2b^2c^2 + 7abc^2 + ab^2c + 12abc$

Exercise $\PageIndex{60}$

$s^{10}(2s^5 + 3s^4 + 4s^3 + 5s^2 + 2s + 2) - s^{15} + 2s^{14} + 3s(s^{12} + 4s^{11}) - s^{10}$

Exercise $\PageIndex{61}$

$6a^4(a^2 + 5)$

Answer

$6a^6 + 30a^4$

Exercise $\PageIndex{62}$

$2x^2y^4(3x^2y + 4xy + 3y)$

Exercise $\PageIndex{63}$

$5m^6(2m^7 + 3m^4 + m^2 + m + 1$

Answer

$10m^{13} + 15m^{10} + 5m^8 + 5m^7 + 5m^6$

Exercise $\PageIndex{64}$

$a^3b^3c^4(4a + 2b + 3c + ab + ac + bc^2$

Exercise $\PageIndex{65}$

$(x+2)(x+3)$

Answer

$x^2 + 5x + 6$

Exercise $\PageIndex{66}$

$(y+4)(y+5)$

Exercise $\PageIndex{67}$

$(a+1)(a+3)$

Answer

$a^2 + 4a + 3$

Exercise $\PageIndex{68}$

$(3x+4)(2x+6)$

Exercise $\PageIndex{69}$

$4xy - 10xy$

Answer

$-6xy$

Exercise $\PageIndex{70}$

$5ab^2 - 3(2ab^2 + 4)$

Exercise $\PageIndex{71}$

$7x^4 - 15x^4$

Answer

$-8x^4$

Exercise $\PageIndex{72}$

$5x^2 + 2x - 3 - 7x^2 - 3x - 4 - 2x^2 - 11$

Exercise $\PageIndex{73}$

$4(x-8)$

Answer

$4x-32$

Exercise $\PageIndex{74}$

$7x(x^2 - x + 3)$

Exercise $\PageIndex{75}$

$-3a(5a - 6)$

Answer

$-15a^2 + 18a$

Exercise $\PageIndex{76}$

$4x^2y^2(2x-3y-5) - 16x^3y^2 - 3x^2y^3$

Exercise $\PageIndex{77}$

$-5y(y^2-3y-6) - 2y(3y^2+7) + (-2)(-5)$

Answer

$-11y^3 + 15y^2 + 16y + 10$

Exercise $\PageIndex{78}$

$-[-(-4)]$

Exercise $\PageIndex{79}$

$−[−(−{−[−(5)]})]$

Answer

$-5$

Exercise $\PageIndex{80}$

$x^2 + 3x - 4 - 4x^2 - 5x - 9 + 2x^2 - 6$

Exercise $\PageIndex{81}$

$4a^2b - 3b^2 - 5b^2 - 8q^2b - 10a^2b - b^2$

Answer

$-6a^2b - 8q^2b - 9b^2$

Exercise $\PageIndex{82}$

$2x^2 - x - (3x^2 - 4x - 5)$

Exercise $\PageIndex{83}$

$3(a−1)−4(a+6)$

Answer

$-a - 27$

Exercise $\PageIndex{84}$

$−6(a+2)−7(a−4)+6(a−1)$

Exercise $\PageIndex{85}$

Add $-3x + 4$ to $5x - 8$.

Answer

$2x - 4$

Exercise $\PageIndex{86}$

Add $4(x^2 - 2x - 3)$ to $-6(x^2 - 5)$.

Exercise $\PageIndex{87}$

Subtract $3$ times $(2x-1)$ from $8$ times $(x-4)$

Answer

$2x - 29$

Exercise $\PageIndex{88}$

$(x+4)(x−6)$

Exercise $\PageIndex{89}$

$(x−3)(x−8)$

Answer

$x^2 - 11x + 24$

Exercise $\PageIndex{90}$

$(2a−5)(5a−1)$

Exercise $\PageIndex{91}$

$(8b+2c)(2b−c)$

Answer

$16b^2 - 4bc - 2c^2$

Exercise $\PageIndex{92}$

$(a-3)^2$

Exercise $\PageIndex{93}$

$(3-a)^2$

Answer

$a^2 - 6a + 9$

Exercise $\PageIndex{94}$

$(x-y)^2$

Exercise $\PageIndex{95}$

$(6x - 4)^2$

Answer

$36x^2 - 48x + 16$

Exercise $\PageIndex{96}$

$(3a-5b)^2$

Exercise $\PageIndex{97}$

$(-x-y)^2$

Answer

$x^2 + 2xy + y^2$

Exercise $\PageIndex{98}$

$(k+6)(k−6)$

Exercise $\PageIndex{99}$

$(m+1)(m−1)$

Answer

$m^2 - 1$

Exercise $\PageIndex{100}$

$(a−2)(a+2)$

Exercise $\PageIndex{101}$

$(3c+10)(3c−10)$

Answer

$9c^2 - 100$

Exercise $\PageIndex{102}$

$(4a+3b)(4a−3b)$

Exercise $\PageIndex{103}$

$(5+2b)(5−2b)$

Answer

$25 - 4b^2$

Exercise $\PageIndex{104}$

$(2y+5)(4y+5)$

Exercise $\PageIndex{105}$

$(y+3a)(2y+a)$

Answer

$2y^2 + 7ay + 3a^2$

Exercise $\PageIndex{106}$

$(6+a)(6−3a)$

Exercise $\PageIndex{107}$

$(x^2 + 2)(x^2 - 3)$

Answer

$x^4 - x^2 - 6$

Exercise $\PageIndex{108}$

$6(a−3)(a+8)$

Exercise $\PageIndex{109}$

$8(2y−4)(3y+8)$

Answer

$48y^2 + 32y - 256$

Exercise $\PageIndex{110}$

$x(x−7)(x+4)$

Exercise $\PageIndex{111}$

$m^2n(m+n)(m+2n)$

Answer

$m^4n + 3m^3n^2 + 2m^2n^3$

Exercise $\PageIndex{112}$

$(b+2)(b^2 - 2b + 3)$

Exercise $\PageIndex{113}$

$3p(p^2 + 5p + 4)(p^2 + 2p + 7)$

Answer

$3p^5 + 21p^4 + 63p^3 + 129p^2 + 84p$

Exercise $\PageIndex{114}$

$(a+6)^2$

Exercise $\PageIndex{115}$

$(x-2)^2$

Answer

$x^2 - 4x + 4$

Exercise $\PageIndex{116}$

$(2x-3)^2$

Exercise $\PageIndex{117}$

$(x^2 + y)^2$

Answer

$x^4 + 2x^2y + y^2$

Exercise $\PageIndex{118}$

$(2m - 5n)^2$

Exercise $\PageIndex{119}$

$(3x^2y^3 - 4x^4y)^2$

Answer

$9x^4y^6 - 24x^6y^4 + 16x^8y^2$

Exercise $\PageIndex{120}$

$(a-2)^4$

Terminology Associated with Equations

Find the domain of the equations for the following problems.

Exercise $\PageIndex{121}$

$y = 8x + 7$

Answer

all real numbers

Exercise $\PageIndex{122}$

$y = 5x^2 - 2x + 6$

Exercise $\PageIndex{123}$

$y = \dfrac{4}{x-2}$

Answer

all real numbers except 2

Exercise $\PageIndex{124}$

$m = \dfrac{-2x}{h}$

Exercise $\PageIndex{125}$

$z = \dfrac{4x+5}{y+10}$

Answer

$x$ can equal any real number; $y$ can equal any number except $-10$