álgebra II semester II
4. Simplify: (2a³)⁻³(3ab²)³
(2⁻³a⁻⁹)(27a³b⁶) 27a³b⁶=27b⁶ ÷8a⁹. ÷8a⁶ 27b⁶/8a^b
10. Rewrite in: a.) standard form b.) vertex form a. y + 3 = 2(x - 1)² b. y=x² + 6x + 3
10. Rewrite in: a.) standard form b.) vertex form a. y + 3 = 2(x - 1)² y+3=2(x-1)(x-1) y+3=2(x²-x-x+1) distribute y+3=2x²-4x+2 -3 -3 -- ---- y=2x²-4x=1 b. y=x² + 6x + 3 y-3=x²+6x -9 -9 --- --- y+6=x²+6x+9 y+6=(x+3)² (6/2)²=(3)²=9 (x+3)(x+3) x²+3x+3x+9 x²+6x+9 y+6=(x+3)²
binomial
2 terms
trinomial
3 terms
3.a. Solve 4m² = 36 /4. /4 √M²=√9 M±3 b. (R - 3)² =16 R+3=±4 4=-3±4 R=1R=-7
3.a. Solve 4m² = 36 /4. /4 √M²=√9 M±3 b. (R - 3)² =16 R+3=±4 4=-3±4 R=1R=-7
factoring TRINOMIALS
3x²-10x-8 =3x+2 times x-4 3x²|-8|=-10x 3x,x|-4,2|-12x+2x=-10x b²-4ac=100+96 =196
4. Solve |2x + 6| = 3
4. Solve |2x + 6| = 3 +/ \- 2X+6=3 2X+6=-3 -6 -6 -6. -6 ---- ----- 2X=-3 2X=-9 /2 /2 /2. /2 X=3/2
1. Expand: (2x + 3y)(4x + 5y)
8x²+10xy+12xy+15y² 8x²+22xy+15y²
if D is a prefecto sq
=factorable
7. Penny invests $1,000 in an account earning 8% interest compounded annually. How much money is in the account after 5 years?
A=10,000(1+.03)¹⁰ $13,439.20
Standard form of a polynomial
AnX^n+An-₁X^n⁻¹+An-2+^n⁻²+...+a₁+d₀ Where n is a non-negative interger and An≠0
ex p.735#10
Asmall∆=1/2bh b=x+y₁n=2+3 Asmall∆=1/2(x+y)(2+3)=1/2(xz+3x+y2+3y) =1/2x2+3/2x+1/2yz+3/2y
discriminant
D = b2 - 4ac determined from the coefficients of the equation ax2 + bx + c = 0.
racional root theorem ex 1 p761 (only list possible 0s,observe graphically
NOT CORRECT f(x)=4x⁴+3x³+4x²+11x+6 =:±1,±2.±3,±6 q:±1,±2,±4 p/q:±1,±1/2,±1/4,±2,±3,±3/2,±3/4,±6 x=1 x=-1 4x³ x1√4x⁴+3x³+4x²+11x+6 -(4x⁴-4x³) 7x³+4x² -(7x³-7x²) ----------- 11x²+11x 1x²-11x ----------- 22x+6 22x-22 --------28
prime polynomials
a polynomial that cant be factored ex. is y²-6 not factorable w/intergers =(x+ sq root sign 6(y-sq root sign 6)bc the dq root of 6 is not an integer PRIME!!
12. Height Formula: h = -16t² + v₀t + h₀ A Ball is thrown upward from an initial height of 4 ft. with an initial velocity of 30 ft/second. a. Find the height of the ball after 1 second. b. At what time will the ball reach its highest point? (sketch a graph to support your answer)
a.h=-16t²+30t+4 h=-16(1)²+30(1)+4 =18ft b/∩<---parabola like that going through the x axis, top of parabola there is a point (0.438,18.1) =.938sec
0 PRODUCT THEOREM
ab=0 if a=0or b=0 ex.find the 0s of: r(X)=(X-5)(2x+3) x-5=0 3x=-3 or x=-3/2 r(x)=0 0=2x³-18x²-20x 0=2x(x²-9x-10 0=2x(x-1)(x+10) 2x=0. x=1 or x=-10
A = P(1 + r)^t.
annual compound interest p=principle r=interest rate t=time(years)
binomials sq factoring
a²+2ab+b²=(a-b)² a²=2ab+b²=(a-b)² ex.factor a)4x²+24x+36 4(x²+6x+9 4(x+3)² b)x²-x+1/4 (x-1/2)² -1(25x²+10x+1) -(5x+1)²
difference of squares
a²-b²=(a+b)(A-b) ex.factor: a0y²-36 =(y+6)(y-6) b064x²-81y² =(8x+9y)(8x-9y) c)x⁴-16 =(x²+4)(x²-4) =(x²+4)(X+2)(X-2) ex.factor: a018x³-8x 2x(3x+2)(3x-2)
A complex number
combination of a real and an imaginary number in the form a + bi a and b are real numbers, and i is the "unit imaginary number" √(−1) The values a and b can be zero. These are all complex numbers: • 1 + i • 2 − 6i • −5.2i (an imaginary number is a complex number with a=0) • 4 (a real number is a complex number with b=0)
# of roots of a polynomial equation theorem
every polynomial equation of degree n has exactly n complex roots(assyming multiple roots are wanted)
fundamental theorem of algebra
every polynomial equation w/ complex numbers of coefficients has at least 1 complex solution
p(x)=x⁴+7x³+2x²-28x-24
ex. a third degree polynomial p(x) has 0s @ 6,1,&2 find 1 possible p(x) 0s:x=6,x=1,x=-2 p(X)=(x-6)(x-1)(x+2) =x³+2x²-7x²-14x+6x+12
the extended distributive property
ex.expand a) (2x-1)(3x²-5x+4) =6x³-10x²+8x-3x²-5x-4 6x³-13x²+13x-4 b)(y²+2y-5) 4y⁴-6y³-y²+8y³-12y²-2y-20y²+30y+5 =4y⁴+2y³-33y²+28y+5
factoring trinomIALS(general)ax2+bx+c might be able to be reqritten int eh form(dx+e)(fx+g) not all trinomials are factorable
ex.factor: a)x²|12|=8x x,x|3,4|3x+4x-7x |2,6|2x+6x=8x |1,12| |-3,-4|
Use a GDC to graph: p(x)=x⁵-4x⁴+x²-5x+50 for -5≤x≤5 and -60≤y≤60
ex.p 734 ex 4 a=12(1.058)⁴+850(1.058)³+975(1.058)²+1175(1.058)+1300 a=$6,144.74
factor theorem:x-r is a factor of p(x) if p(r)=0
ex.pg.755 ex 3 0s:x=-6,x=-2,x=-1,x=2 | | ∨ p(x)=(x+6)(x+2)(x+1)(x-2) (x²+8x+12)(x²-x-2) x⁴-x³-2x²+8x³-8x²-16x+12x²-12x-24
monomial factoring
factor a monomial out of multiply terms. ex.factor a)12x²-4x =4x(3x-1) b)15x³y+5x²y²-35xy² =5xy(3x²+xy-7y)
A = P(1 + r/n)^nt
general compound interest n=#times compounded/year
degree of polynomial(in more than one variable):The largest sum of the degrees of the variables for any of the terms
idk
racional root theorem
ifAnX^n+An-₁X^n⁻¹+An-2+^n⁻²+...+a₁+a₀ then any rational 0s of f(x) Will be p/q p=factor of a₀ q=factor of An
polynomials are classified by the number of terms
monomial:1 term
sketch the graph of y=x^n when:
n is odd(pic) n is even-A v shape that is symmetrical, above the x axis
Important for rational root theorem
only rational 0s can be found using the rational root theorem. any additional 0s should be irrational or complex numbers
# of roots of a polynomial equation theorem p768 ex
p(x)=x⁵-x⁴-21-+x³-37x²=98x-24 then graph cross x axis=|R(real) 0s =3 imaginar y=2/5=remainder
degree of a polynomial
the largest exponent of the variable Ex.write in standard form(5x³-6)² (5x³-6)²=25x⁶-60x³+36 Ex.if p(x)=x⁵-4x+x²-5x+50 find p(-1) p(-1)=(-1)⁵-4(-1)⁴+(-1)²+5(-1)+50 =-1-4+1+5+50=51
5. Use the quadratic formula to solve (give exact answers, not approximations): 2x² + -3x -3 = 0
x=-b±√b²-4ac/2a x=4±√40/4 x=4±2√5/4 x=2±√5/2
3. A basketball player shoots from a free throw line that is 15' from the center of the circular rim which is 10' off the ground at the point (0,10). The player releases the ball from 6' above the ground at the point (15,6). The ball passes through the point (7,14). a. Sketch the graph of the path of the ball on the graph: b. Use the graphing calculator to find a quadratic regression, round each value to 3 decimal places. c. Predict the height of the ball when it is 3 ft. away from the center of the rim
∩<--graph like that but with a shorter initial point like 4 when the end point is like 0 for the x value and a point in the middle at the top b)y=0.19x²+1,452x c)≈13ft
