Pre Calculus 12 BC Test Prep.

Factor by grouping to obtain the factors 3x+2 and 5x^2-4, then set each factor to zero. Solve: x = -2/3, and x = ± sqrt(4/5) as the roots.

Factor the trinomial by choosing numbers that multiply to 30 and sum to -13. Apply the difference of squares to obtain (x+2)(x-2)(x+3)(x-3) and solve for x to find x-intercepts.

By testing possible factors, x-1 is a factor; synthetic division yields x^2-3x-10, which factors to (x-5)(x+2); thus the polynomial factors as (x-1)(x-5)(x+2) with roots 1, 5, -2.

perform polynomial long division of the expression by 3x+1, writing the coefficients (including 0x); obtain quotient 2x^2 with remainder -5 and verify by multiplication.

Apply the remainder theorem to a polynomial, solve for zeros, evaluate P(-1), and conclude that X+1 is a factor with zero remainder.

Apply the factor theorem by evaluating at x = -2 to ensure x+2 is a factor of the polynomial; this gives k = 34.

Solve for k by substituting x = -2 into p(x) = 4x^2 + 2kx - 5, since the remainder is 3 when divided by x+2, giving k = 2.

Solve for remainder when X is divided by 5x minus 3 by substituting x = 3/5 and computing the resulting value to obtain 496/125.

Evaluate the polynomial at x = 5 to perform division by x minus 5 and compute the result, which is 768.

Examine the detailed solution for solving 3x−1=0, deducing x=1/3 and showing that substituting x=1/3 into this function of x yields a remainder of zero.

Apply the rational root theorem to find possible zeroes by testing ± factors of the constant term over the leading coefficient; for 10 and 1, test ±1, ±2, ±5, ±10.

Identify zeros from factors of the constant term, test candidates to find x-1 as a factor, then factor the quadratic to (x+2)(x-5) and solve for x (-2, 1, 5).

Identify roots at x = -1 and x = 2, use synthetic division to factor the polynomial, and conclude with (x+1)(x-2)(2x+3)(x+2) for a complete solution.

Learn how the leading coefficient determines end behavior and the overall shape of polynomial graphs, with examples of even versus odd degree, x-intercepts, and y-intercepts.

present a detailed solution to question 20 on a degree-4 polynomial with zeros at -1 and 4, finding the y-intercept and sketching the graph with a positive leading coefficient.

Derive the equation from the graph by using the x-intercepts at 1 and -2, express f(x) as a(x+1)(x-2), and use the y-intercept to solve for a.

Use the degree of a polynomial to determine the maximum number of x intercepts. Here the degree is five, so up to five x intercepts are possible.

Factor and test candidate linear factors to match x squared terms and coefficients, then verify by expansion to confirm the correct answer.

Factor the rational function, identify holes and asymptotes, and plot intercepts to sketch the graph, noting the vertical asymptote at 4/3 and the hole at 3 and 1/5.

learn to sketch a rational function by identifying its vertical asymptote at x = -3, the horizontal asymptote, and its x-intercept and y-intercept, including x = 6/5.

Identify the graph restrictions and nonpermissible x-values in this pre calculus 12 bc test prep, then analyze why the horizontal asymptote arises from numerator degree one over denominator degree two.

Demonstrates solving a rational expression with radicals by multiplying by the conjugate, applying the difference of squares, and expanding to simplify to a final fraction; identifies option C.

Solve a division problem with a common base by converting x squared to x^(1/2), subtracting exponents to x^(3/2), then convert to radical form for the final answer.

Apply long division to simplify the function, identify the oblique asymptote y = x + 5 and vertical asymptote x = 4, and compute the y-intercept (0, 9/4) for sketching.

Analyze the rational function (x^2-25)/(x-1) by identifying vertical asymptote x=1, slant asymptote y=x+1, x-intercepts x=±5, and y-intercept 25, then sketch accordingly.

Demonstrates a detailed solution using foil to multiply binomials with minus three and three, showing how to handle multiplication steps, squares, and sign changes to reach the final answer.

Determine the domain of the composite function x = a x g(x) by enforcing a nonnegative radicand (x-2 ≥ 0), and applying correct inequality rules when multiplying by -1.

Use elimination by adding the two equations to cancel terms, then apply substitution to solve for x and verify the solution.

Detailed solution to question 14 shows solving for c and e by combining negative and positive terms, handling brackets, and confirming the sum equals eight to reach the final answer.

Use a base-case count to determine placements, obtain 16 possibilities, replace elements to reach 20, and compute 400 total, yielding option c.

Demonstrate the detailed solution to question 17 by converting expressions to fractions, handling negative values, and multiplying to simplify toward the final result.

This detailed solution for question 18 walks through computing f∘g(-5) by expanding and foiling polynomials, distributing terms, and combining like terms to reach the final result.

Substitute x with g(x) and simplify to form the quadratic x^2 + 12x + 27, confirming option B as the equation.

Learn how to find the inverse of a function by completing the square, switching x and y, and solving for y in terms of x.

Find the inverse by swapping coordinates; swap x and y for each point, turning (3,9) into (9,3) and (-1,-5) into (-5,-1).

Learn to find the inverse of a linear function by swapping x and y, solving for y. Use cross-multiplication and proper notation to demonstrate the inverse function.

Determine the inverse of a quadratic by selecting three points, swapping x and y to form the inverse coordinates, and sketching them on the graph to visualize the relation.

Explore coterminal angles and convert between degrees and radians, using 360 degrees equals 2π and adding or subtracting multiples of 2π to find equivalent angles.

Convert the rotation fractions to degrees by multiplying by 360, using the calculator to compute (-13/18 + 11/18) × 360 and obtain the negative degree result.

Convert a negative degree angle to rotations by dividing by 360, showing -4000 degrees equals -11 1/9 rotations, and express 0.111 repeating as 1/9 to finalize the result.

Convert 5π/3 radians to degrees using the 180 degrees per pi ratio, showing how 5π/3 corresponds to 300 degrees.

Convert from radian to degree using 180/π, simplify the expression, and compute the angle as 270 degrees.

Convert degrees to radians by flipping the ratio 180 degrees over pi, then simplify by cross-canceling common factors to obtain pi over 3 for the given angle.

Convert 3.8 radians to degrees by multiplying by 180/pi, and round to the nearest degree to obtain 218 degrees.

Apply the arc length formula s = r theta in radians to compute arc length; with angle 1.64 rad and radius 20 m, arc length equals 32.8 m.

Compute the arc length by multiplying the radius 5.1 by the angle pi radians, yielding about 16.02.

Apply the arc length formula s = r theta to solve for the radius. With theta as 1 radian and the arc length as 5, obtain r = 5.

Calculate the swing's arc length using s = r theta for a 4-meter radius and theta = pi/16. The result is s = pi/4 meters.

Explains identifying an angle's measure and converting degrees to radians using pi over 180, highlighting memorization of radian values and noting 3 pi over 2 as a key example.

determine sine of angle in quadrant 2 from tan equals -2/5, identify opposite 2 and adjacent 5, compute hypotenuse sqrt29, and express sine as 2 sqrt29 over 29.

Determine the x coordinate of point p on the unit circle (radius 1) in the second quadrant using similar triangles and proportional reasoning to obtain x = -2√13/13.

Analyze how a transformed cosine graph determines solution counts on the given interval. Show how compressing by 1/5 creates five periods, yielding ten solutions.

Determine the number of solutions by analyzing the graph and its asymptotes within a transformed interval. Use the transformed graph to count solutions and verify with a graphing calculator.

This lecture provides the detailed solution to question 25 by applying the formula for the number of solutions and examining the untransformed form and its transformation.

An analysis of a trig function with period 3, phase shift 3/2, and vertical shift 3 shows how to form the expression using pi/3 x and a plus 3.

Solve question 30 by converting the period to degrees, showing it equals 180 degrees, and selecting option B.

Determine the period of a trigonometric function by drawing the graph, locating the midline, marking key points, and counting squares tied to pi fractions.

Identify the middle line at negative 2, then compute the amplitude as half the vertical distance, and conclude that the answer is C.

This lecture solves question 35 by identifying amplitude, midline, period, and phase shift for a cosine model, then uses the starting point and maximum to confirm the final answer.

From the graph, set amplitude to 25 and midline to 30, use a cosine with period 12, yielding y = 25 cos(π/6 x) + 30 with no phase shift.

Identify which two trigonometric functions share the same period, compute the period, and conclude that functions a and b have the same period, answering question 37.

Identify the range by using amplitude: from B − |A| to B + |A|, with maximum at B + |A| and minimum at B − |A|.

Determine the sine graph with amplitude 3 and middle line -2, giving max 1 and min -5, and identify the period and key points on the grid.

Apply the box method to sketch two periods of the sine graph, locate the minimum and key points, and plot the period-ending points to construct the graph.

Apply the ball's method to sketch a sine wave with midline 3 and amplitude 0.5, max 3.5 and min 2.5. Plot one and two periods on the grid.

The lecture shows solving a calculator-based trigonometry problem in radian mode by substituting 143/365 into an expression with pi and cos, then converting hours to minutes to obtain the time.

The lecture solves question 45 by graphing a trigonometric function, using a calculator to locate intercepts within a 40-second interval, and estimating when the function is below 0.1.

Students use a graphing calculator in radian mode to locate the intersection of the functions via tracing, adjust the window for clarity, and obtain the intersection at about 1.375.

Learn factoring to solve trigonometric equations by hand, then apply cosine with 60-30-90 triangles, determine principal and general solutions, and identify the cosine period.

Determine quadrant iv from positive cosine and negative sine, and compute the angle's coordinates: (4, -3) with hypotenuse 5, yielding cos = 4/5 and sin = -3/5.

the lecture shows how to find the general solution to a trig problem using substitution and back-substitution, solving for multiple angle cases and expressing all solutions with pi multiples.

Solve question 10 by determining exact angle measures, using 45-45-90 triangle relationships, quadrant restrictions, and sign conventions to derive 135°, −45°, and the remaining 45° angles.

Solve trigonometric equations using unit circle reasoning and reciprocal forms, obtaining general solutions; x = 7π/6 + 2πk and x = 11π/6 + 2πk.

Set sin x = a, deduce a = -1/2, use the 30-degree reference angle, and find x in [0, 2π): 7π/6 and 11π/6.

Solve a trigonometric equation within the domain using a 30-60-90 triangle to obtain sin x = sqrt(3)/2, and identify x = pi/3 and x = 2pi/3.

Solve question 15 by squaring to get sin^2 x = 1/2, using 30-60-90 and 45-45-90 triangles, reference angle pi/4, and list x in radians: pi/4, 3pi/4, 5pi/4, 7pi/4.

Explains why sine squared plus cosine squared equals one by writing tangent as sine over cosine, and cotangent as cosine over sine, then combining to a common denominator.

Pre calculus 12 bc test prep lecture explains simplifying trigonometric expressions with identities like sine squared plus cosine squared equals one to identify equivalent forms.

Compute question 20 by substituting values and applying reciprocals in the trig expression, leading to cotangent and the final answer a.

Explore a step-by-step trigonometric simplification, rewriting cos^2 x in terms of sin^2 x using sin^2 x + cos^2 x = 1, and identifying the correct simplified form.

Rewrite all expressions in terms of sine and cosine, identify reciprocals such as secant and cosecant, and determine the correct angle relationships.

Simplify the expression by converting everything to sine and cosine, use a common denominator, and apply sin^2 x + cos^2 x = 1 to arrive at 1/cos x.

Derive the general radian solution of a trig equation by identifying sine and cosine constraints, solving for key values, and generalizing to all x using pi intervals.

Provide a detailed solution for proving a trig identity in pre calculus 12 bc test prep, using sin^2 x + cos^2 x = 1 to simplify and verify both sides.

Apply double-angle identities to evaluate cos(2x) using 1 minus 2 sin^2 x and cos^2 x minus sin^2 x, with x = pi/6, confirming cos(pi/3) = 1/2.

Apply algebraic manipulation of trig terms to rewrite the expression as sin^2 θ minus cos^2 θ, leading to the conclusion that the correct choice is B.

Learn how to prove the identity by cross multiplying, using conjugates, and applying the difference of squares to simplify expressions and verify both sides are equal.

Turn everything to sine and cosine, rewrite as one big square root, and simplify using sin^2 x + cos^2 x = 1 to obtain cos^2 x over sin^2 x.

Use conjugates to form a common denominator, then apply sin^2 x = 1 − cos^2 x to prove the left and right sides of the identity match.

solve for x using sine and cosine relationships and cross-multiplication, identify solutions from 0 to 2π, and use 30-60-90 triangles for angles where sin x is 0 or 1/2.

Recognize the sine addition identity sin A cos B + cos A sin B and identify the result sin(A+B).

Sketch an exponential function by tabling x and y for y = (1/2)^x + 1 - 6, identify horizontal asymptote y = -6, and locate x- and y-intercepts.

Learn to solve this equation by making the bases equal, applying exponent laws, cross-canceling, and isolating x to reach the solution.

Equate bases by rewriting 32 as 2^5 and 8 as 2^3, then solve the resulting linear equation to find x values, giving x = 2 and x = -4.

Learn to solve exponential equations by rewriting bases as powers of three, equating exponents, and solving for x, including negative exponents and power rules.

Solve for x in an exponential equation with base 7, using 7^3 = 343 and simplifying to determine x equals -6.

This solution uses log laws to rewrite the function, builds a table of points, identifies the domain x>1 with a vertical asymptote at x=1, and sketches the curve.

solve a logarithmic equation by enforcing the log’s domain (argument > 0), applying change-of-base, aligning bases, moving coefficients, and isolating x to obtain the solution.

Apply exponent rules to cancel exponents, cross out terms, and raise both sides to a power to simplify and evaluate the expression.

Question 16 applies a new rule to form a single law when bases match, rewriting additions by multiplying, crossing out common bases, and estimating the result near 40.

Apply logarithm laws to convert subtraction into division and combine expressions into a single log, then solve for x in question 17 of pre calculus 12 bc test prep.

This detailed solution to question 18 rewrites the expression as a product, factors to x-1 and x-6, checks positivity constraints (x>5), and verifies the solution x=6.

Walks through solving a precalculus question by multiplying, rearranging terms, and determining domain restrictions to identify valid x values.

Apply the laws of logarithms to simplify and evaluate a log expression on a calculator, using brackets and division to arrive at four.

Apply logarithm properties to rewrite the expression as the ratio of logs, cross terms, and reveal the final result: log base a of b.

Apply exponent laws to simplify equations by crossing out identical factors on both sides, compare base and exponent relationships, and verify the result; the method shown is preferred.

The lecture analyzes question 26, applying a rule to cross out terms, presents x in Lexington form, and shows two equivalent expressions using Beatty's.

The lecture explains rewriting question 30 in terms of law a and law b using separate law and the OVP rule. It shows that the final answer is b.

Apply logarithm laws and exponential form to simplify and solve a complex expression, converting to base 10 and isolating terms to reach the solution for question 31.

Applies log rules to rewrite a fraction as differences of logarithms, breaks numbers into factors such as eight times eight times nine, and simplifies to an expression like a minus two c minus two b.

Apply exponent rules to simplify expressions by combining exponents, then solve x^3 = 27 to find x = 3.

Solve for x using the steps shown: move the exponent to the coefficient, expand and combine terms, then divide to isolate x in this pre calculus 12 bc course.

Provide a detailed solution to question 36 from pre calculus 12 bc test prep, guiding students to move terms, apply exponent and log rules, and isolate x by division.

This detailed solution uses the half-life model to find the time for a 200 g substance to decay to 140 g, applying N = N0 (1/2)^(t/T) and solving for t.

Explore the time to decay from 230 g to 15 g using an exponential decay model, a half-life of 7.4 days, and change-of-base logs.

Model direct sunlight brightness as it passes through six sheets of glass, with each sheet transmitting 10 percent; set A to 100 percent and define x as t over T.

Solve a bacteria growth problem by applying exponential growth with a six-fold increase every 24 hours. Compute the 48-hour population as 2000 times 6 squared, giving 72,000.

Demonstrates solving an exponential scale problem by comparing two scales, using a calculator to compute the power ratio and determine how many times one scale is bigger.

Solve a factorial ratio using cross-multiplication, isolate a, and verify with a calculator that the expression equals 6720 in this question 2 solution for pre calculus 12 bc test prep.

This lecture presents a detailed solution to question 3 using n choose r and factorials, showing step-by-step simplification, cross-multiplication, and verifying that the correct option is B.

Detail a solution by evaluating a decreasing product with factorial concepts. Start with a middle guess like 10, then test 6 and 5 using a calculator to identify correct choice.

The lecture demonstrates using a calculator to test numbers, trying 13 and 12 times 13 to see if the result is 169, and states that the final answer is c.

Demonstrate how to simplify a factorial ratio by canceling terms and flipping expressions, showing manual calculation when a calculator cannot handle large numbers, with final result 1.

Explains the handshake problem using combinations, showing three people yield three handshakes and the general formula n choose 2 applies to eleven people, yielding answer B.

Determine a six-character license plate with two letters followed by four digits from 1 to 9. Multiply the choices to obtain the total possibilities: 26^2 times 9^4.

For a 10-question multiple-choice test with 5 options per question, there are 5^10 possible answer sheets. The caption states the correct choice is C.

Determine the total options by multiplying four car choices, five truck choices, and six motorcycle choices to find the overall count of possibilities.

determine how many ways eight color-coded wires with eight colors can be arranged, multiplying the choices for each position to yield 40320 possible combinations.

Learn how to count circular arrangements: for 14 scouts around a table, the number of ways equals 13!, illustrating circular permutation.

Compute the number of ways to distribute identical items—pencils, three rulers, two notebooks, and a pen—among ten students using factorials with repetition, yielding 12,600 ways.

explains counting permutations of the word Manhattan by accounting for repeated letters, using nine letters with three a's, two n's, and two t's, dividing by 3! 2! 2!.

Count six-digit numbers greater than 700000 formed from digits 2, 2, 4, 4, 4, and 7 with the first digit 7, giving 10.

Calculate ways to assign three offices from 14 students with at least two girls. Subtract cases with no girls or only one girl from 14p3 to confirm 826.

Using permutations, choose 2 vowels from 3 and 2 consonants from 4, then arrange the 4 selected letters; this yields 3×6×24=432, i.e., option c.

In this question, the coach selects five members from ten and designates a captain, showing that 10 choose 5 times 5 equals 1260, so the answer is option B.

The solution uses the counting principle: 14 choices for captain and 13 for co-captain, yielding 182 ways.

Compute the number of ways to seat nine people in four chairs by using 9p4, showing 9×8×7×6 equals 3024 and that order matters.

Calculate the number of two or more card combinations from a 16-card subset of a 52-card deck. Apply binomial coefficients (n choose k) to count red and black card selections.

Learn to count all possible choices by multiplying group-level options with individual-level options, using a setup with two groups, five options, and three selections.

Explore permutations to create unique quiz versions by ordering matters, showing that 5P5 yields 120 and 6P6 yields 720, ensuring at least 128 distinct versions for 126 students.

Calculate the total number of possible bingo cards from column ranges, then convert a one-card-per-second count into years to illustrate the enormous timespan.

Apply binomial expansion using the binomial formula to determine coefficients. Select k and compute combinations to find the resulting term.

the lecture walks through binomial expansion by expanding (x+y)^2 and (x+y)^3, revealing the terms x^2+2xy+y^2 and x^3+3x^2y+3xy^2+y^3. it explains how higher powers produce more terms and the pattern of coefficients.

Identify the middle term in a 13-term expansion and compute the seventh term using the combination formula.

Determine the constant term in the binomial expansion by equating exponents: -30+2k+k=0, giving k=10, which corresponds to the 11th term.

Apply binomial expansion techniques to solve question 39, using binomial coefficients such as 7 choose 4 and 7 choose 7 to identify terms and derive the final expression.

Learn to read Pascal's triangle and convert a specific entry to binomial notation, illustrated by finding the 7th entry in the 9th row as 8 choose 6.

Explore the detailed solution to question 42 on binomial expansions, comparing coefficients using binomial coefficients and k-value selections to determine which expansion yields a larger coefficient.

Count the paths from point a to point b by moving right and down, shifting numbers to the right or down and adding corners to reach the final total.