11th grade algebra

Example

Simplify: (5x² + 4x − 9) − (−5x² − 2x + 1)

1. 10x² + 6x − 10

Answer: 10x^2+6x-10

Example

Find the remainder when x³ − 3x² − 7x − 6 is divided by (x + 1).

1. f(−1) = −1 − 3 + 7 − 6
2. = −3

Answer: -3

Example

Simplify (5 + 5i) + (8 − 7i). Write it as a + bi.

1. 13 − 2i

Answer: 13-2i

Example

A population of 1000 grows by 5% each year. Write the model P(t) = 1000(b)ᵗ and find P(5), to the nearest whole number.

1. P(t) = 1000(1.05)ᵗ
2. P(5) ≈ 1276.28
3. ≈ 1276

Answer: 1276

Example

Evaluate log_5(1/5).

1. 5⁻¹ = 1/5
2. log_5(1/5) = −1

Answer: -1

Example

Solve: √(x − 10) = 2

1. x − 10 = 4
2. x = 14

Answer: x = 14

Example

Solve: −66/x − 5 = 6

1. −66/x = 11
2. −66 = 11x
3. x = −6

Answer: x = -6

Example

Which transformation maps y = f(x) to y = f(x + 2)?

1. translate left 2

Answer: translate left 2

Example

Find the sum of the first 28 terms of the arithmetic series −7 + (−11) + (−15) + …

1. S = 28/2 × (2 × (−7) + 27 × (−4))
2. = 14 × (−122) = −1708

Answer: -1708

Example

What is the minimum value of f(x) = (x − 3)² + 6?

1. At x = 3 the bracket is 0, so f = 6.

Answer: 6

Example

Expand and simplify: (x + 4)(x² − 4x − 9)

1. x(x² − 4x − 9) + 4(x² − 4x − 9)
2. x³ − 25x − 36

Answer: x^3-25x-36

Example

Expand and simplify: (x − 4)(x + 4)(x − 2)

1. (x − 4)(x + 4) = x² − 16
2. (x² − 16)(x − 2)
3. x³ − 2x² − 16x + 32

Answer: x^3-2x^2-16x+32

Example

Factorise: x³ + 8

1. (x + 2)(x² − 2x + 4)

Answer: (x + 2)(x² − 2x + 4)

Example

Divide x³ + 7x² + 18x + 24 by (x + 4). The remainder is 0. Type the quotient.

1. x³ + 7x² + 18x + 24 = (x + 4)(x² + 3x + 6)
2. x² + 3x + 6

Answer: x^2+3x+6

Example

(x − 2) is a factor of x³ + kx² − 3x − 10. Find k.

1. 2³ + k × 2² − 3 × 2 − 10 = 0
2. 4k − 8 = 0
3. k = 2

Answer: 2

Example

x = −5 is a solution of x³ + 3x² − 13x − 15 = 0. Find all the solutions, separated by commas.

1. (x + 5)(x² − 2x − 3) = 0
2. (x + 5)(x − 3)(x + 1) = 0
3. x = −5, −1, 3

Answer: x = -5 or x = -1 or x = 3

Example

Write as a single fraction: 4/(x − 2) + 4/(x + 1)

1. (4(x + 1) + 4(x − 2))/((x − 2)(x + 1))
2. (8x − 4)/((x − 2)(x + 1))

Answer: (8x − 4)/((x − 2)(x + 1))

Example

Evaluate log₂ 32.

1. 32 = 2^5
2. log₂ 32 = 5

Answer: 5

Example

Write log₁₀ 1000 = 3 in exponential form.

1. 10^3 = 1000

Answer: 10^3 = 1000

Example

log 8 + log 9 − log 8 = log N. Find N.

1. log(8 × 9 ÷ 8)
2. N = 9

Answer: 9

Example

Evaluate log₃ 3 + log₃ 3.

1. log₃ 9
2. = 2

Answer: 2

Example

a = log x, b = log y and c = log z. Write log(x⁴y/z²) in terms of a, b and c.

1. 4a + b − 2c

Answer: 4a + b − 2c

Example

Solve 3^(x + 3) = 27.

1. 27 = 3^3
2. x + 3 = 3
3. x = 0

Answer: x = 0

Example

Solve 4^x = 122. Give x to 3 decimal places.

1. x = log 122 ÷ log 4
2. x ≈ 3.465

Answer: 3.465

Example

Solve log₁₀(2x) = 2.

1. 2x = 100
2. x = 50

Answer: x = 50

Example

An arithmetic series has 15 terms. The first term is 8 and the last term is 92. Find the sum.

1. 15 × (8 + 92) ÷ 2 = 750

Answer: 750

Example

Find the sum of the first 6 terms of 1 + 3 + 9 + …

1. 1(3^6 − 1)/(3 − 1)
2. = 364

Answer: 364

Example

u₁ = −3 and uₙ₊₁ = −uₙ + 0. Find u₄.

1. u₂ = 3, u₃ = −3
2. u₄ = 3

Answer: 3

Example

Write a quadratic x² + bx + c with roots −5 and −7. Type it.

1. (x + 5)(x + 7)
2. x² + 12x + 35

Answer: x^2+12x+35

Example

Write y = x² − 2x − 8 in vertex form.

1. (x − 1)² = x² − 2x + 1
2. y = (x − 1)² − 9

Answer: y = (x − 1)² − 9

Example

Which transformation maps y = f(x) onto y = −f(x)?

1. Reflection in the x-axis

Answer: Reflection in the x-axis

Example

f(x) = x² − 4 when x < −1, and f(x) = −2x + 3 when x ≥ −1. Find f(−2).

1. −2 < −1: use x² − 4
2. f(−2) = 0

Answer: 0

Example

Solve: |x + 5| ≥ 3

1. x ≤ −8 or x ≥ −2

Answer: x ≤ −8 or x ≥ −2