4.10: Exercise Supplement
Algebraic Expressions
For the following problems, write the number of terms that appear, then write the terms.
\(4x^2 + 7x + 12\)
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three: \(4x^2, 7x, 12\)
\(14y^6\)
\(c + 8\)
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two: \(c, 8\)
\(8\)
List, if any should appear, the common factors for the following problems.
\(a^2 + 4a^2 + 6a^2\)
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\(a^2\)
\(9y^4 - 18y^4\)
\(12x^2y^3 + 36y^3\)
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\(12y^3\)
\(6(a+4) + 12(a+4)\)
\(4(a+2b)+6(a+2b)\)
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\(2(a+2b)\)
\(17x^2y(z+4) + 51y(z+4)\)
\(6a^2b^3c + 5x^2y\)
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no common factors
For the following problems, answer the question of how many.
\(x\)'s in \(9x\)?
\((a+b)\)'s in \(12(a+b)\)?
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12
\(a^4\)'s in \(6a^4\)
\(c^3\)'s in \(2a^2bc^3\)?
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\(2a^2b\)
\((2x+3y)^2\)'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.
\(8z, z\)
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\(8\)
\(16a^3b^2c^4, c^4\)
\(7y(y+3), 7y\)
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\((y+3)\)
\((-5)a^5b^5c^5, bc\)
Equations
For the following problems, observe the equations and write the relationship being expressed.
\(a = 3b\)
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The value of \(a\) is equal to three times the value of \(b\).
\(r = 4t + 11\)
\(f = \dfrac{1}{2}m^2 + 6g\)
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The value of \(f\) is equal to six times \(g\) more then one half times the value of \(m\) squared.
\(x = 5y^3 + 2y + 6\)
\(P^2 = ka^3\)
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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.
\(C = 2 \pi r\). Find \(C\) is \(\pi\) is approximated by \(3.14\) and \(r = 6\)
\(I = \dfrac{E}{R}\). Find \(I\) is \(E = 20\) and \(R = 2\).
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\(10\)
\(I=prt\). Find \(I\) if \(p=1000\), \(r=0.06\), and \(t=3\).
\(E = mc^2\). Find \(E\) if \(m = 120\) and \(c = 186,000\).
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\(4.1515 \times 10^{12}\)
\(z = \dfrac{x-u}{s}\). Find \(z\) if \(x = 42\), \(u = 30\), and \(s = 12\).
\(R = \dfrac{24C}{P(n+1)}\). Find \(R\) if \(C = 35\), \(P = 300\), and \(n = 19\).
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\(\dfrac{7}{50}\) 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.
\(2a+9\)
\(4y^3 + 3y + 1\)
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trinomial, cubic; 4, 3, 1
\(10a^4\)
\(147\)
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monomial; zero; 147
\(4xy + 2yz^2 + 6x\)
\(9ab^2c^2 + 10a^3b^2c^5\)
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binomial; tenth; 9, 10
\((2xy^3)^0, xy^3 \not = 0\)
Why is the expression \(\dfrac{4x}{3x-7}\) not a polynomial?
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... because there is a variable in the denominator
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.
\(3y + 2x = 1\)
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linear
\(4a^2 - 5a + 8 = 0\)
\(y - x - z + 4w = 21\)
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linear
\(5x^2 + 2x^2 - 3x + 1 = 19\)
\((6x^3)^0 + 5x^2 = 7\)
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Quadratic
Combining Polynomials Using Addition and Subtraction- Special Binomial Products
Simplify the algebraic expressions for the following problems.
\(4a^2b + 8a^2b - a^2b\)
\(21x^2y^3 + 3xy + x^2y^3 + 6\)
- Answer
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\(22x^2y^3 + 3xy + 6\)
\(7(x+1)+2x−6\)
\(2(3y^2+4y+4)+5y^2+3(10y+2)\)
- Answer
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\(11y^2 + 38y + 14\)
\(5[3x + 7(2x^2 + 3x + 2) + 5] - 10x^2 + 4(3x^2 + x)\)
\(8{3[4y^3+y+2] + 6(y^3+2y^2)} - 24y^3 - 10y^2 - 3\)
- Answer
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\(120y^3 + 86y^2 + 24y + 45\)
\(4a^2bc^3 + 5abc^3 + 9abc^3 + 7a^2bc^2\)
\(x(2x+5) + 3x^2 - 3x + 3\)
- Answer
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\(5x^2 + 2x + 3\)
\(4k(3k^2 + 2k + 6) + k(5k^2 + k) + 16\)
\(2{5[6(b+2a+c^2)]}\)
- Answer
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\(60c^2 + 120a + 60b\)
\(9x^2y(3xy + 4x) - 7x^3y^2 - 30x^3y + 5y(x^3y + 2x)\)
\(3m[5 + 2m(m+6m^2)] + m(m^2 + 4m + 1)\)
- Answer
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\(36m^4 + 7m^3 + 4m^2 + 16m\)
\(2r[4(r + 5) - 2r - 10] + 6r(r + 2)\)
\(abc(3abc + c + b) + 6a(2bc + bc^2)\)
- Answer
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\(3a^2b^2c^2 + 7abc^2 + ab^2c + 12abc\)
\(s^{10}(2s^5 + 3s^4 + 4s^3 + 5s^2 + 2s + 2) - s^{15} + 2s^{14} + 3s(s^{12} + 4s^{11}) - s^{10}\)
\(6a^4(a^2 + 5)\)
- Answer
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\(6a^6 + 30a^4\)
\(2x^2y^4(3x^2y + 4xy + 3y)\)
\(5m^6(2m^7 + 3m^4 + m^2 + m + 1\)
- Answer
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\(10m^{13} + 15m^{10} + 5m^8 + 5m^7 + 5m^6\)
\(a^3b^3c^4(4a + 2b + 3c + ab + ac + bc^2\)
\((x+2)(x+3)\)
- Answer
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\(x^2 + 5x + 6\)
\((y+4)(y+5)\)
\((a+1)(a+3)\)
- Answer
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\(a^2 + 4a + 3\)
\((3x+4)(2x+6)\)
\(4xy - 10xy\)
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\(-6xy\)
\(5ab^2 - 3(2ab^2 + 4)\)
\(7x^4 - 15x^4\)
- Answer
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\(-8x^4\)
\(5x^2 + 2x - 3 - 7x^2 - 3x - 4 - 2x^2 - 11\)
\(4(x-8)\)
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\(4x-32\)
\(7x(x^2 - x + 3)\)
\(-3a(5a - 6)\)
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\(-15a^2 + 18a\)
\(4x^2y^2(2x-3y-5) - 16x^3y^2 - 3x^2y^3\)
\(-5y(y^2-3y-6) - 2y(3y^2+7) + (-2)(-5)\)
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\(-11y^3 + 15y^2 + 16y + 10\)
\(-[-(-4)]\)
\(−[−(−{−[−(5)]})]\)
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\(-5\)
\(x^2 + 3x - 4 - 4x^2 - 5x - 9 + 2x^2 - 6\)
\(4a^2b - 3b^2 - 5b^2 - 8q^2b - 10a^2b - b^2\)
- Answer
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\(-6a^2b - 8q^2b - 9b^2\)
\(2x^2 - x - (3x^2 - 4x - 5)\)
\(3(a−1)−4(a+6)\)
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\(-a - 27\)
\(−6(a+2)−7(a−4)+6(a−1)\)
Add \(-3x + 4\) to \(5x - 8\).
- Answer
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\(2x - 4\)
Add \(4(x^2 - 2x - 3)\) to \(-6(x^2 - 5)\).
Subtract \(3\) times \((2x-1)\) from \(8\) times \((x-4)\)
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\(2x - 29\)
\((x+4)(x−6)\)
\((x−3)(x−8)\)
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\(x^2 - 11x + 24\)
\((2a−5)(5a−1)\)
\((8b+2c)(2b−c)\)
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\(16b^2 - 4bc - 2c^2\)
\((a-3)^2\)
\((3-a)^2\)
- Answer
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\(a^2 - 6a + 9\)
\((x-y)^2\)
\((6x - 4)^2\)
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\(36x^2 - 48x + 16\)
\((3a-5b)^2\)
\((-x-y)^2\)
- Answer
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\(x^2 + 2xy + y^2\)
\((k+6)(k−6)\)
\((m+1)(m−1)\)
- Answer
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\(m^2 - 1\)
\((a−2)(a+2)\)
\((3c+10)(3c−10)\)
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\(9c^2 - 100\)
\((4a+3b)(4a−3b)\)
\((5+2b)(5−2b)\)
- Answer
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\(25 - 4b^2\)
\((2y+5)(4y+5)\)
\((y+3a)(2y+a)\)
- Answer
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\(2y^2 + 7ay + 3a^2\)
\((6+a)(6−3a)\)
\((x^2 + 2)(x^2 - 3)\)
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\(x^4 - x^2 - 6\)
\(6(a−3)(a+8)\)
\(8(2y−4)(3y+8)\)
- Answer
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\(48y^2 + 32y - 256\)
\(x(x−7)(x+4)\)
\(m^2n(m+n)(m+2n)\)
- Answer
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\(m^4n + 3m^3n^2 + 2m^2n^3\)
\((b+2)(b^2 - 2b + 3)\)
\(3p(p^2 + 5p + 4)(p^2 + 2p + 7)\)
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\(3p^5 + 21p^4 + 63p^3 + 129p^2 + 84p\)
\((a+6)^2\)
\((x-2)^2\)
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\(x^2 - 4x + 4\)
\((2x-3)^2\)
\((x^2 + y)^2\)
- Answer
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\(x^4 + 2x^2y + y^2\)
\((2m - 5n)^2\)
\((3x^2y^3 - 4x^4y)^2\)
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\(9x^4y^6 - 24x^6y^4 + 16x^8y^2\)
\((a-2)^4\)
Terminology Associated with Equations
Find the domain of the equations for the following problems.
\(y = 8x + 7\)
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all real numbers
\(y = 5x^2 - 2x + 6\)
\(y = \dfrac{4}{x-2}\)
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all real numbers except 2
\(m = \dfrac{-2x}{h}\)
\(z = \dfrac{4x+5}{y+10}\)
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\(x\) can equal any real number; \(y\) can equal any number except \(-10\)