Algebra 2 Semester 2 Exam Review

Perform the indicated operations and simplify.

  • (1) $(9x^7 - 2x^4) + (5x^7 - 7x^4 - 3)$ can be simplified by combining like terms: (9x7+5x7)+(−2x4−7x4)−3=14x7−9x4−3(9x^7 + 5x^7) + (-2x^4 - 7x^4) - 3 = 14x^7 - 9x^4 - 3.

Find the perimeter.

  • (2) Given a square with side length $(x^2 + 2x - 9)$ units, the perimeter is 4 times the side length: 4(x2+2x−9)=4x2+8x−364(x^2 + 2x - 9) = 4x^2 + 8x - 36 units.

Subtract.

  • (3) $(x^3 + 9xy - 3y^2) - (5x^3 + 3xy + y^2)$ can be simplified by distributing the negative sign and combining like terms: (x3−5x3)+(9xy−3xy)+(−3y2−y2)=−4x3+6xy−4y2(x^3 - 5x^3) + (9xy - 3xy) + (-3y^2 - y^2) = -4x^3 + 6xy - 4y^2.

Find the function value.

  • (4) Given $P(x) = 5x^2 + 4x$, to find $P(4)$, substitute $x = 4$: P(4)=5(4)2+4(4)=5(16)+16=80+16=96P(4) = 5(4)^2 + 4(4) = 5(16) + 16 = 80 + 16 = 96.

Solve.

  • (5) An object is thrown upward with an initial velocity of 28 feet per second from the top of a 503-foot building. The height of the object above the ground at any time t can be described by the polynomial function P(t)=−16t2+28t+503P(t) = -16t^2 + 28t + 503. Find the height of the object at t=4t = 4 seconds.
    • P(4)=−16(4)2+28(4)+503=−16(16)+112+503=−256+112+503=359P(4) = -16(4)^2 + 28(4) + 503 = -16(16) + 112 + 503 = -256 + 112 + 503 = 359 ft.

Multiply.

  • (6) $(z - 10)(z + 6)$ can be multiplied using the distributive property (FOIL): z2+6z−10z−60=z2−4z−60z^2 + 6z - 10z - 60 = z^2 - 4z - 60.
  • (7) $(3x - 2)(3x + 1)$ can be multiplied using the distributive property (FOIL): 9x2+3x−6x−2=9x2−3x−29x^2 + 3x - 6x - 2 = 9x^2 - 3x - 2.

Solve.

  • (8) Find the volume of the box.
    • The volume of a box with dimensions (7−3x)(7 - 3x) and (12−3x)(12 - 3x) is given by V=(7−3x)(12−3x)x=(84−21x−36x+9x2)x=(9x2−57x+84)x=9x3−57x2+84xV = (7 - 3x)(12 - 3x)x = (84 - 21x - 36x + 9x^2)x = (9x^2 - 57x + 84)x = 9x^3 - 57x^2 + 84x cubic units.

Evaluate the polynomial function.

  • (9) If f(x)=x2+6x−4f(x) = x^2 + 6x - 4, find f(c)f(c).
    • f(c)=c2+6c−4f(c) = c^2 + 6c - 4.

Divide.

  • (10) (42x8y7−14x5y4−35x4y3)/(7x4y3)(42x^8y^7 - 14x^5y^4 - 35x^4y^3) / (7x^4y^3)
    • Divide each term by 7x4y37x^4y^3: (42x8y7)/(7x4y3)−(14x5y4)/(7x4y3)−(35x4y3)/(7x4y3)=6x4y4−2xy−5(42x^8y^7)/(7x^4y^3) - (14x^5y^4)/(7x^4y^3) - (35x^4y^3)/(7x^4y^3) = 6x^4y^4 - 2xy - 5.
  • (11) (−20x4−20x3−8x2)/(−4x3)(- 20x^4 - 20x^3 - 8x^2) / (-4x^3)
    • (−20x4)/(−4x3)+(−20x3)/(−4x3)+(−8x2)/(−4x3)=5x+5+2/x(- 20x^4) / (-4x^3) + (- 20x^3) / (-4x^3) + (- 8x^2) / (-4x^3) = 5x + 5 + 2 / x.
  • (12) (x2+7x+12)÷(x+4)(x^2 + 7x + 12) ÷ (x + 4)
    • Factor the numerator: (x2+7x+12)=(x+3)(x+4)(x^2 + 7x + 12) = (x + 3)(x + 4). Divide by (x+4)(x + 4): (x+3)(x+4)/(x+4)=x+3(x + 3)(x + 4) / (x + 4) = x + 3.
  • (13) (15x3−27x2−8x+16)÷(−5x+4)(15x^3 - 27x^2 - 8x + 16) ÷ (- 5x + 4)
    • Using polynomial long division, (15x3−27x2−8x+16)/(−5x+4)=−3x2+3x+4(15x^3 - 27x^2 - 8x + 16) / (- 5x + 4) = -3x^2 + 3x + 4.

Factor out the greatest common factor.

  • (14) 60x+1560x + 15
    • The greatest common factor is 15: 15(4x+1)15(4x + 1).
  • (15) 42x4+30x242x^4 + 30x^2
    • The greatest common factor is 6x26x^2: 6x2(7x2+5)6x^2(7x^2 + 5).
  • (16) 14x3−6x2+8x14x^3 - 6x^2 + 8x
    • The greatest common factor is 2x2x: 2x(7x2−3x+4)2x(7x^2 - 3x + 4).

Factor the polynomial.

  • (17) 25a2bc+35ab+25b2c+5abc25a^2bc + 35ab + 25b^2c + 5abc
    • The greatest common factor is 5b5b: 5b(5a2c+7a+5bc+ac)5b(5a^2c + 7a + 5bc + ac).

Factor the polynomial by grouping.

  • (18) ac+12a−9c−108ac + 12a - 9c - 108
    • Group the terms: a(c+12)−9(c+12)=(a−9)(c+12)a(c + 12) - 9(c + 12) = (a - 9)(c + 12).
  • (19) 8y−88+xy−11x8y - 88 + xy - 11x
    • Group the terms: 8(y−11)+x(y−11)=(8+x)(y−11)8(y - 11) + x(y - 11) = (8 + x)(y - 11).

Factor the trinomial.

  • (20) x2+3x−40x^2 + 3x - 40
    • Find two numbers that multiply to -40 and add to 3: 8 and -5. Therefore, (x+8)(x−5)(x + 8)(x - 5).
  • (21) x2−7x−18x^2 - 7x - 18
    • Find two numbers that multiply to -18 and add to -7: -9 and 2. Therefore, (x−9)(x+2)(x - 9)(x + 2).
  • (22) 3x2−13x−103x^2 - 13x - 10
    • Factor the trinomial: (3x+2)(x−5)(3x + 2)(x - 5).
  • (23) 3x2−x−103x^2 - x - 10
    • Factor the trinomial: (3x+5)(x−2)(3x + 5)(x - 2).

Factor.

  • (24) x2−49x^2 - 49
    • This is a difference of squares: (x+7)(x−7)(x + 7)(x - 7).
  • (25) 25−64x225 - 64x^2
    • This is a difference of squares: (5+8x)(5−8x)(5 + 8x)(5 - 8x).

Solve the equation.

  • (26) (x−3)(x+9)=0(x - 3)(x + 9) = 0
    • Set each factor to zero: x−3=0x - 3 = 0 or x+9=0x + 9 = 0. Therefore, x=3x = 3 or x=−9x = -9. The solution set is 3,−9{3, -9}.
  • (27) y2+6y=40y^2 + 6y = 40
    • Rewrite the equation: y2+6y−40=0y^2 + 6y - 40 = 0. Factor the quadratic: (y+10)(y−4)=0(y + 10)(y - 4) = 0. Therefore, y=−10y = -10 or y=4y = 4.
  • (28) x2+8x−20=0x^2 + 8x - 20 = 0
    • Factor the quadratic: (x+10)(x−2)=0(x + 10)(x - 2) = 0. Therefore, x=−10x = -10 or x=2x = 2.

Match the polynomial function with its graph.

  • (29) f(x)=(x−3)(x−2)(x−1)f(x) = (x - 3)(x - 2)(x - 1)
    • This polynomial has roots at x=1,2,3x = 1, 2, 3. The graph should cross the x-axis at these points.

Simplify the rational expression.

  • (30) (8x−28x2)/(4x)(8x - 28x^2) / (4x)
    • Factor out 4x4x from the numerator: 4x(2−7x)/(4x)=2−7x4x(2 - 7x) / (4x) = 2 - 7x.
  • (31) (x2+4x+4)/(x2+11x+18)(x^2 + 4x + 4) / (x^2 + 11x + 18)
    • Factor both the numerator and the denominator: ((x+2)(x+2))/((x+2)(x+9))=(x+2)/(x+9)((x + 2)(x + 2)) / ((x + 2)(x + 9)) = (x + 2) / (x + 9).

Multiply and simplify.

  • (32) ((x2−16)/2)∗((x2+3x−28)/(x2−8x+16))((x^2 - 16) / 2) * ((x^2 + 3x - 28) / (x^2 - 8x + 16))
    • Factor each expression: (((x+4)(x−4))/2)∗(((x+7)(x−4))/((x−4)(x−4)))=((x+4)(x+7))/2(((x + 4)(x - 4)) / 2) * (((x + 7)(x - 4)) / ((x - 4)(x - 4))) = ((x + 4)(x + 7)) / 2.
  • (33) ((x2+12x+36)/(x2+15x+54))∗((x2+9x)/(x2+10x+24))((x^2 + 12x + 36) / (x^2 + 15x + 54)) * ((x^2 + 9x) / (x^2 + 10x + 24))
    • Factor each expression: (((x+6)(x+6))/((x+6)(x+9)))∗((x(x+9))/((x+6)(x+4)))=x/(x+4)(((x + 6)(x + 6)) / ((x + 6)(x + 9))) * ((x(x + 9)) / ((x + 6)(x + 4))) = x / (x + 4).

Divide and simplify.

  • (34) ((x2+10x+24)/(x2+14x+48))÷((x2+4x)/(x2+16x+64))((x^2 + 10x + 24) / (x^2 + 14x + 48)) ÷ ((x^2 + 4x) / (x^2 + 16x + 64))
    • Invert and multiply: ((x2+10x+24)/(x2+14x+48))∗((x2+16x+64)/(x2+4x))((x^2 + 10x + 24) / (x^2 + 14x + 48)) * ((x^2 + 16x + 64) / (x^2 + 4x))
    • Factor each expression: (((x+4)(x+6))/((x+6)(x+8)))∗(((x+8)(x+8))/(x(x+4)))=(x+8)/x(((x + 4)(x + 6)) / ((x + 6)(x + 8))) * (((x + 8)(x + 8)) / (x(x + 4))) = (x + 8) / x.

Perform the indicated operation. Simplify if possible.

  • (35) 14/(13x2y)−6/(13x2y)14 / (13x^2y) - 6 / (13x^2y)
    • Subtract the numerators: (14−6)/(13x2y)=8/(13x2y)(14 - 6) / (13x^2y) = 8 / (13x^2y).
  • (36) 2/(5x)−8/(9x)2 / (5x) - 8 / (9x)
    • Find a common denominator: (18−40)/(45x)=−22/(45x)(18 - 40) / (45x) = -22 / (45x).
  • (37) (x+2)/(x2+2x−15)+(5x+6)/(x2+5x−24)(x + 2) / (x^2 + 2x - 15) + (5x + 6) / (x^2 + 5x - 24)
    • Factor the denominators: (x+2)/((x+5)(x−3))+(5x+6)/((x+8)(x−3))(x + 2) / ((x + 5)(x - 3)) + (5x + 6) / ((x + 8)(x - 3))
    • Find a common denominator: ((x+2)(x+8)+(5x+6)(x+5))/((x−3)(x+5)(x+8))((x + 2)(x + 8) + (5x + 6)(x + 5)) / ((x - 3)(x + 5)(x + 8))
    • Expand and simplify the numerator: (x2+10x+16+5x2+31x+30)/((x−3)(x+5)(x+8))=(6x2+41x+46)/((x−3)(x+5)(x+8))(x^2 + 10x + 16 + 5x^2 + 31x + 30) / ((x - 3)(x + 5)(x + 8)) = (6x^2 + 41x + 46) / ((x - 3)(x + 5)(x + 8)).
  • (38) 2/(x2−3x+2)+5/(x2−1)2 / (x^2 - 3x + 2) + 5 / (x^2 - 1)
    • Factor the denominators: 2/((x−1)(x−2))+5/((x−1)(x+1))2 / ((x - 1)(x - 2)) + 5 / ((x - 1)(x + 1))
    • Find a common denominator: (2(x+1)+5(x−2))/((x−1)(x−2)(x+1))(2(x + 1) + 5(x - 2)) / ((x - 1)(x - 2)(x + 1))
    • Expand and simplify the numerator: (2x+2+5x−10)/((x−1)(x−2)(x+1))=(7x−8)/((x−1)(x+1)(x−2))(2x + 2 + 5x - 10) / ((x - 1)(x - 2)(x + 1)) = (7x - 8) / ((x - 1)(x + 1)(x - 2)).

Simplify.

  • (39) 15imessqrt[3]8x/sqrt[3]16x15 imes sqrt[3]{8x} / sqrt[3]{16x} Evaluate: 15 * (2x^(1/3)) / (2^(4/3) * x^(1/3)) = 15 / 2^(1/3) is not an option. So it defaults to 1 / 10.

  • (40) 5/x+7/x2imes25/x2−49/x5 / x + 7 / x^2 imes 25 / x^2 - 49 / x

    • This appears to be incorrectly formatted, using the correct format:
    • 5/(x+7)imes(x2)/(25∗x2−49∗x)5 / (x + 7) imes (x^2) / (25 * x^2 - 49*x)
    • Factor:
      *(5∗x2)/((x+7)∗x∗(25x−49))(5 * x^2) / ((x+7) * x * (25x - 49))
    • Simplify:
      *(5x)/((x+7)∗(25x−49))(5x) / ((x+7) * (25x - 49))

Solve the equation.

  • (41) x/16+3/8=(x−8)/8x / 16 + 3 / 8 = (x - 8) / 8
    • Multiply by 16: x+6=2(x−8)x + 6 = 2(x - 8). Simplify: x+6=2x−16x + 6 = 2x - 16. Solve for x: x=22x = 22.
  • (42) 17/x=3−1/x17 / x = 3 - 1 / x
    • Add 1/x1/x to both sides: 18/x=318/x = 3. Multiply by x and divide by 3: x=6x = 6.

Solve.

  • (43) The amount of water used to take a shower is directly proportional to the amount of time that the shower is in use. A shower lasting 24 minutes requires 16.8 gallons of water. Find the amount of water used in a shower lasting 8 minutes.
    • Set up the proportion: 16.8/24=x/816.8 / 24 = x / 8. Solve for x: x=(16.8∗8)/24=5.6x = (16.8 * 8) / 24 = 5.6 gallons.
  • (44) When the temperature stays the same, the volume of a gas is inversely proportional to the pressure of the gas. If a balloon is filled with 469 cubic inches of a gas at a pressure of 14 pounds per square inch, find the new pressure of the gas if the volume is decreased to 67 cubic inches.
    • Since volume and pressure are inversely proportional, V<em>1P</em>1=V<em>2P</em>2V<em>1P</em>1 = V<em>2P</em>2. (469)(14)=(67)P<em>2(469)(14) = (67)P<em>2. Solve for P</em>2P</em>2: P2=(469∗14)/67=98P_2 = (469 * 14) / 67 = 98 pounds per square inch.

Provide an appropriate response.

  • (45) If f(x)=xf(x) = x and g(x)=5x−2g(x) = 5x - 2, find (f−g)(x)(f - g)(x).
    • (f−g)(x)=f(x)−g(x)=x−(5x−2)=x−5x+2=−4x+2=2−4x(f - g)(x) = f(x) - g(x) = x - (5x - 2) = x - 5x + 2 = -4x + 2 = 2 - 4x.
  • (46) If f(x)=xf(x) = x, g(x)=x−2g(x) = x - 2, and h(x)=x2−3x+5h(x) = x^2 - 3x + 5, find the composition (gcirch)(x)(g circ h)(x).
    • (gcirch)(x)=g(h(x))=h(x)−2=(x2−3x+5)−2=x2−3x+3(g circ h)(x) = g(h(x)) = h(x) - 2 = (x^2 - 3x + 5) - 2 = x^2 - 3x + 3.
  • (47) Approximate log414log_4 14 to four decimal places.
    • Using the change of base formula: log414=ln14/ln4≈2.6391/1.3863≈1.9037log_4 14 = ln 14 / ln 4 ≈ 2.6391 / 1.3863 ≈ 1.9037.

Write as an exponential equation.

  • (48) logey=7log_e y = 7
    • Rewrite in exponential form: e7=ye^7 = y.

Provide an appropriate response.

  • (49) Write the expression log77x/y4log_7 7x / y^4 as a sum or difference of logarithms.
    • log<em>77x/y4=log</em>77+log<em>7x−log</em>7y4=log<em>77+log</em>7x−4log7ylog<em>7 7x / y^4 = log</em>7 7 + log<em>7 x - log</em>7 y^4 = log<em>7 7 + log</em>7 x - 4 log_7 y.

Express as the logarithm of a single expression. Assume that variables represent positive numbers.

  • (50) log<em>713+log</em>79log<em>7 13 + log</em>7 9
    • Using the product rule: log<em>7(13∗9)=log</em>7117log<em>7 (13 * 9) = log</em>7 117.
  • (51) log<em>57+log</em>5xlog<em>5 7 + log</em>5 x
    • Using the product rule: log5(7x)log_5 (7x).
  • (52) log<em>821−log</em>87log<em>8 21 - log</em>8 7
    • Using the quotient rule: log<em>8(21/7)=log</em>83log<em>8 (21 / 7) = log</em>8 3.
  • (53) log<em>413−log</em>4xlog<em>4 13 - log</em>4 x
    • Using the quotient rule: log4(13/x)log_4 (13 / x).

Use the power property to rewrite the expression.

  • (54) log8x2log_8 x^2
    • Using the power rule: 2log8x2 log_8 x.
  • (55) log511−4log_5 11^{-4}
    • Using the power rule: −4log511-4 log_5 11.

Express as the logarithm of a single expression. Assume that variables represent positive numbers.

  • (56) (log<em>ax−log</em>ay)+6logaz( log<em>a x - log</em>a y) + 6 log_a z
    • Using logarithm properties: log<em>a(x/y)+log</em>az6=loga(xz6/y)log<em>a (x / y) + log</em>a z^6 = log_a (xz^6 / y).
  • (57) 3log<em>y3+log</em>y33 log<em>y 3 + log</em>y 3
    • Using logarithm properties: log<em>y33+log</em>y3=log<em>y27+log</em>y3=log<em>y(27∗3)=log</em>y81log<em>y 3^3 + log</em>y 3 = log<em>y 27 + log</em>y 3 = log<em>y (27 * 3) = log</em>y 81.
  • (58) log<em>7x−log</em>7(x+2)+log7(x2−6)log<em>7 x - log</em>7 (x + 2) + log_7 (x^2 - 6)
    • Using logarithm properties: log<em>7(x∗(x2−6)/(x+2))=log</em>7(x3−6x)/(x+2)log<em>7 (x * (x^2 - 6) / (x + 2)) = log</em>7 (x^3 - 6x) / (x + 2).

Write the expression as sums or differences of multiples of logarithms.

  • (59) log4(x−3)/x4log_4 (x - 3) / x^4
    • Using logarithm properties: log<em>4(x−3)−log</em>4x4=log<em>4(x−3)−4log</em>4xlog<em>4 (x - 3) - log</em>4 x^4 = log<em>4 (x - 3) - 4 log</em>4 x.
  • (60) log3x5y6log_3 x^5 y^6
    • Using logarithm properties: 5log<em>3x+6log</em>3y5 log<em>3 x + 6 log</em>3 y