Algebra Concepts and Expressions Review

Solving for a Variable

10 A piece of wire of length

66 cm

is bent to form the five sides of a pentagon. The pentagon consists of three sides of a rectangle and two sides of an equilateral triangle. The sides of the rectangle measure

x cm

and

y cm

and the sides of the triangle measure

x cm

, as shown in the diagram below. 10 (a) (i) You are given that

\sin 60^{\circ} = \frac{\sqrt{3}}{2}

Explain why the area of the triangle is

\frac{\sqrt{3}}{4} x^{2}

[1 mark] 10 (a) (ii) Show that the area enclosed by the wire,

A {cm}^{2}

, can be expressed by the formula

A = 33 x - \frac{1}{4} (6 - \sqrt{3}) x^{2}

[3 marks]

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Polynomial Division

Given the polynomial

P (x) = 2 x^{3} - 3 x^{2} + 4 x - 5

and

Q (x) = x - 1

, find the remainder when

P (x)

is divided by

Q (x)

. Additionally, if

R (x)

is the remainder and

θ

is the root of

Q (x)

, find the value of

\cos (θ)

R (θ) = \cos (θ)

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Partial Fraction Decomposition

Solve for

x

in the equation

\frac{\cos (x)}{\cos (x) - \sin (x)} = 2

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Factoring Using Any Method

Solve the following trigonometric equation for

x

2 \sin (x) \cos (x) + \sin (x) - 1 = 0

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Factoring a Difference of Squares

\sin^{2} x - \cos^{2} x = 1

, find the value of

x

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Factoring a Sum of Cubes

Find the value of

x

\cos (2 x) = \frac{1}{2}

and

x^{3} + 1 = 0

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Factoring Out Greatest Common Factor (GCF)

Given two algebraic expressions

4 x^{2} \cos^{2} y + 9 x^{2} \sin^{2} y

and

5 x^{2} \cos^{2} y + 2 x^{2} \sin^{2} y

, find the greatest common factor (GCF) and simplify the expressions.

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Factoring by Grouping

\sin (x) = a

and

\cos (x) = b

, express the expression

2 \sin (x) \cos (x) - 2 a^{2} + 2 b^{2}

in its simplest form.

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Factoring Trinomials

Solve the equation

2 \sin^{2} x - 3 \sin x - 2 = 0

in the interval

[0, 2 π)

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Finding the Domain

Find the domain of the function

y = \sqrt{\sin (x)} + \frac{1}{x - 3}

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Dividing

x = \frac{π}{6}

, find the value of

\frac{2 c o s x + s i n x}{s i n x}

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Evaluate the Expression Using the Given Values

Evaluate the expression

2 c o s (x) + 3 s i n (y)

given that

c o s (x) = \frac{1}{2}

and

s i n (y) = \frac{3}{5}

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Evaluating Radicals

Given that

x = \sqrt[3]{27}

and

y = \cos (\frac{π}{3})

, solve for

z = 3 x^{2} - 2 y

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Finding the Holes in a Graph

Find the holes in the graph of the function

f (x) = \frac{\sin x - 1}{x^{2} - 1}

More examples

Solving for a Variable

Polynomial Division

Partial Fraction Decomposition

Factoring Using Any Method

Factoring a Difference of Squares

Factoring a Sum of Cubes

Factoring Out Greatest Common Factor (GCF)

Factoring by Grouping

Factoring Trinomials

Finding the Domain

Dividing

Evaluate the Expression Using the Given Values

Evaluating Radicals

Finding the Holes in a Graph

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