\((3d + 2)(3d - 2)(9d^2 + 4) = \dotso\)
A. \(9d^4 - 4\)
B. \(9d^4 + 16\)
C. \(81d^4 - 16\)
D. \(81d^4 + 16\)
\((3d + 2)(3d - 2)(9d^2 + 4) = \dotso\)
A. \(9d^4 - 4\)
B. \(9d^4 + 16\)
C. \(81d^4 - 16\)
D. \(81d^4 + 16\)
Solution: C
\begin{equation*} \begin{split} (3d + 2)(3d - 2)(9d^2 + 4)& = [(3d)^2 - 2^2][(3d)^2 + 4]\\\\ (3d + 2)(3d - 2)(9d^2 + 4)& = (9d^2 - 4)(9d^2 + 4)\\\\ (3d + 2)(3d - 2)(9d^2 + 4)& = (9d^2)^2 - 4^2\\\\ (3d + 2)(3d - 2)(9d^2 + 4)& = 81d^4 - 16 \end{split} \end{equation*}
Factorize \(25s^2 + 4t^2 + 49u^2 - 20st - 70su + 28tu\)
A. \((5s - 2t - 7u)^2\)
B. \((5s - 2t +7u)^2\)
C. \((5s - 3t - 7u)^2\)
D. \((7s - 4t - 7u)^2\)
Factorize \(25s^2 + 4t^2 + 49u^2 - 20st - 70su + 28tu\)
A. \((5s - 2t - 7u)^2\)
B. \((5s - 2t +7u)^2\)
C. \((5s - 3t - 7u)^2\)
D. \((7s - 4t - 7u)^2\)
Solution: A
\(\color{blue}1^{st}\) step: root each element which has power of two
\begin{equation*} \begin{split} 25s^2 &\rightarrow \sqrt{(5s)^2} = \color{purple}(5s) \color{black}\text{ or } (-5s)\\\\ 4t^2 &\rightarrow \sqrt{(2t)^2} = (2t) \text{ or } \color{purple}(-2t)\\\\ 49u^2 &\rightarrow \sqrt{(7u)^2} = (7u)\text{ or }\color{purple}(-7u) \end{split} \end{equation*}
\(\color{blue}2^{nd}\) step: look at the two times of each two elements
\begin{equation*} \begin{split} -20st &\rightarrow 2(5s)(-2t)\\\\ -70su &\rightarrow 2(5s)(-7u)\\\\ 28tu &\rightarrow 2(-2t)(-7u) \end{split} \end{equation*}
\(\color{blue}3^{rd}\) step: \(25s^2 + 4t^2 + 49u^2 - 20st - 70su + 28tu = (5s - 2t - 7u)^2\)
\((2x - 5z)(4x^2 + 10xz + 25z^2) = \dotso\)
A. \(8x^3 - 125z^3\)
B. \(8x^3 + 125z^3\)
C. \(8x^3 - 95z^3\)
D. \(8x^3 - 125z^3\)
\((2x - 5z)(4x^2 + 10xz + 25z^2) = \dotso\)
A. \(8x^3 - 125z^3\)
B. \(8x^3 + 125z^3\)
C. \(8x^3 - 95z^3\)
D. \(8x^3 - 125z^3\)
Solution: D
\begin{equation*} \begin{split} (2x - 5z)(4x^2 + 10xz + 25z^2)& = (2x - 5z)((2x)^2 + 2x\cdot 5z + (5z)^2)\\\\ (2x - 5z)(4x^2 + 10xz + 25z^2)& = (2x)^3 - (5z)^3\\\\ (2x - 5z)(4x^2 + 10xz + 25z^2)& = 8x^3 - 125z^3 \end{split} \end{equation*}
\((3p - 2q)^2 - (3p + q)^2 = \dotso\)
(A) \(3p \: (p - 6q)\)
(B) \(3p \: (6p - q)\)
(C) \(3q \: (6q - p)\)
(D) \(3q \: (q - 6p)\)
\((3p - 2q)^2 - (3p + q)^2 = \dotso\)
(A) \(3p \: (p - 6q)\)
(B) \(3p \: (6p - q)\)
(C) \(3q \: (6q - p)\)
(D) \(3q \: (q - 6p)\)
Solution: D
\begin{equation*} \begin{split} & (3p)^2 - 2 \:.\: 3p \:.\: 2q + (2q)^2 - \left[(3p)^2 + 2 \:.\: 3p \:.\: q + q^2 \right]\\\\ & 9p^2 - 12pq + 4q^2 - \left[9p^2 + 6pq + q^2 \right]\\\\ & \cancel{9p^2} - 12pq + 4q^2 - \cancel{9p^2} - 6pq - q^2\\\\ & -18pq + 3q^2 \end{split} \end{equation*}
Simplify \(\dfrac{9-9x+2x^2}{15-14x+3x^2}\)
A. \(\dfrac{2-3x}{5+3x}\)
B. \(\dfrac{3-2x}{5-3x}\)
C. \(\dfrac{3+2x}{5-3x}\)
D. \(\dfrac{5-2x}{5+3x}\)
Simplify \(\dfrac{9-9x+2x^2}{15-14x+3x^2}\)
A. \(\dfrac{2-3x}{5+3x}\)
B. \(\dfrac{3-2x}{5-3x}\)
C. \(\dfrac{3+2x}{5-3x}\)
D. \(\dfrac{5-2x}{5+3x}\)
Solution: B
\begin{equation*} \begin{split} \frac{9-9x+2x^2}{15-14x+3x^2}& = \frac{(3-2x)\cancel{(3-x)}}{(5-3x)\cancel{(3-x)}}\\\\ \frac{9-9x+2x^2}{15-14x+3x^2}& = \frac{3-2x}{5-3x} \end{split} \end{equation*}
Simplify \(\dfrac{x^3-y^3}{x^2 -3xy + 2y^2}\)
A. \(\dfrac{x^2 - xy + y^2}{x-y}\)
B. \(\dfrac{x^2 + xy - y^2}{2x-y}\)
C. \(\dfrac{x^2 + xy + y^2}{x-y}\)
D. \(\dfrac{x^2 + xy + y^2}{x-2y}\)
Simplify \(\dfrac{x^3-y^3}{x^2 -3xy + 2y^2}\)
A. \(\dfrac{x^2 - xy + y^2}{x-y}\)
B. \(\dfrac{x^2 + xy - y^2}{2x-y}\)
C. \(\dfrac{x^2 + xy + y^2}{x-y}\)
D. \(\dfrac{x^2 + xy + y^2}{x-2y}\)
Solution: D
\begin{equation*} \begin{split} \frac{x^3-y^3}{x^2 -3xy + 2y^2}& = \frac{\cancel{(x-y)}(x^2+xy + y^2)}{\cancel{(x-y)}(x-2y)}\\\\ \frac{x^3-y^3}{x^2 -3xy + 2y^2}& = \frac{x^2 + xy + y^2}{x-2y} \end{split} \end{equation*}
Simplify \(\dfrac{(x^2 - 3)(2x - 5)}{x + \sqrt{3}} \times \dfrac{2}{2x^2 -27x + 55}\)
A. \(\dfrac{2x - \sqrt{3}}{x + 7}\)
B. \(\dfrac{2x + 2\sqrt{3}}{x - 9}\)
C. \(\dfrac{2x - 2\sqrt{3}}{x - 11}\)
D. \(\dfrac{2x - 2\sqrt{3}}{x + 11}\)
Simplify \(\dfrac{(x^2 - 3)(2x - 5)}{x + \sqrt{3}} \times \dfrac{2}{2x^2 -27x + 55}\)
A. \(\dfrac{2x - \sqrt{3}}{x + 7}\)
B. \(\dfrac{2x + 2\sqrt{3}}{x - 9}\)
C. \(\dfrac{2x - 2\sqrt{3}}{x - 11}\)
D. \(\dfrac{2x - 2\sqrt{3}}{x + 11}\)
Solution: C
\begin{equation*} \begin{split} &\frac{(x^2 - 3)(2x - 5)}{x + \sqrt{3}} \times \frac{2}{2x^2 -27x + 55}\\\\ &\frac{[x^2 - (\sqrt{3})^2](2x - 5)}{x + \sqrt{3}} \times \frac{2}{(2x-5)(x - 11)}\\\\ &\frac{\cancel{(x + \sqrt{3})}(x - \sqrt{3})\cancel{(2x - 5)}}{\cancel{x + \sqrt{3}}} \times \frac{2}{\cancel{(2x-5)}(x - 11)}\\\\ &\frac{2x - 2\sqrt{3}}{x - 11} \end{split} \end{equation*}
Simplify \(\dfrac{x-2y}{2}-\dfrac{x-y}{6} + \dfrac{3x + y}{3}\)
A. \(\dfrac{8x-3y}{6}\)
B. \(\dfrac{8x+ 2y}{6}\)
C. \(\dfrac{8x-3y}{8}\)
D. \(\dfrac{8x-5y}{12}\)
Simplify \(\dfrac{x-2y}{2}-\dfrac{x-y}{6} + \dfrac{3x + y}{3}\)
A. \(\dfrac{8x-3y}{6}\)
B. \(\dfrac{8x+ 2y}{6}\)
C. \(\dfrac{8x-3y}{8}\)
D. \(\dfrac{8x-5y}{12}\)
Solution: A
\begin{equation*} \begin{split} &\frac{3(x-2y)}{6}-\dfrac{x-y}{6}+ \frac{2(3x + y)}{6}\:\:\:\:\:\color {blue} \text{the least common denominator is 6}\\\\ &\frac{3(x-2y)-(x-y)+2(3x + y)}{6}\\\\ &\frac{3x-6y-x+y + 6x + 2y}{6}\\\\ &\frac{3x-x + 6x-6y+y + 2y}{6}\\\\ &\frac{8x-3y}{6} \end{split} \end{equation*}
Simplify \(\dfrac{4y}{y^2+3y-10}+\dfrac{3}{y-2}\)
A. \(\dfrac{7y + 10}{(y+1)(y-2)}\)
B. \(\dfrac{7y - 15}{(y+3)(y-2)}\)
C. \(\dfrac{7y + 15}{(y+5)(y-2)}\)
D. \(\dfrac{9y + 20}{(y+7)(y-2)}\)
Simplify \(\dfrac{4y}{y^2+3y-10}+\dfrac{3}{y-2}\)
A. \(\dfrac{7y + 10}{(y+1)(y-2)}\)
B. \(\dfrac{7y - 15}{(y+3)(y-2)}\)
C. \(\dfrac{7y + 15}{(y+5)(y-2)}\)
D. \(\dfrac{9y + 20}{(y+7)(y-2)}\)
Solution: C
\begin{equation*} \begin{split} &\frac{4y}{(y+5)(y-2)}+\frac{3}{y-2}\:\:\:\:\:\color {blue} \text{the least common denominator is }(y + 5)(y - 2)\\\\ &\frac{4y}{(y+5)(y-2)}+\frac{3(y+5)}{(y+5)(y-2)}\\\\ &\frac{4y}{(y+5)(y-2)}+\frac{3(y+5)}{(y+5)(y-2)}\\\\ &\frac{4y+3(y+5)}{(y+5)(y-2)}\\\\ &\frac{4y + 3y + 15}{(y+5)(y-2)}\\\\ &\frac{7y + 15}{(y+5)(y-2)} \end{split} \end{equation*}
\(\dfrac{10x}{x^2+3x-4}-\dfrac{2}{x-1} = \dotso\)
A. \(\dfrac{2}{x+4}\)
B. \(\dfrac{4}{x+4}\)
C. \(\dfrac{6}{x+4}\)
D. \(\dfrac{8}{x+4}\)
\(\dfrac{10x}{x^2+3x-4}-\dfrac{2}{x-1} = \dotso\)
A. \(\dfrac{2}{x+4}\)
B. \(\dfrac{4}{x+4}\)
C. \(\dfrac{6}{x+4}\)
D. \(\dfrac{8}{x+4}\)
Solution: D
The simple form of \(\dfrac{6x^2 + 31x + 35}{3x^2 - 28x - 55}\) is ...
A. \(\dfrac{3x+5}{x - 11}\)
B. \(\dfrac{2x + 7}{3x + 5}\)
C. \(\dfrac{2x + 7}{x - 11}\)
D. \(\dfrac{2x-7}{x + 11}\)
The simple form of \(\dfrac{6x^2 + 31x + 35}{3x^2 - 28x - 55}\) is ...
A. \(\dfrac{3x+5}{x - 11}\)
B. \(\dfrac{2x + 7}{3x + 5}\)
C. \(\dfrac{2x + 7}{x - 11}\)
D. \(\dfrac{2x-7}{x + 11}\)
Solution: C
\begin{equation*} \begin{split} \frac{6x^2 + 31x + 35}{3x^2 - 28x - 55}& = \frac{\cancel{(3x + 5)}(2x + 7)}{\cancel{(3x + 5)}(x - 11)}\\\\ \frac{6x^2 + 31x + 35}{3x^2 - 28x - 55}& = \frac{2x + 7}{x - 11} \end{split} \end{equation*}
The length of a rectangle is \(2 \text{ cm}\) less than two times the width. If the area is \(112 \text{ cm}^2\), the length of the rectangle is ...
A. \(11 \text{ cm}\)
B. \(12 \text{ cm}\)
C. \(13 \text{ cm}\)
D. \(14 \text{ cm}\)
The length of a rectangle is \(2 \text{ cm}\) less than two times the width. If the area is \(112 \text{ cm}^2\), the length of the rectangle is ...
A. \(11 \text{ cm}\)
B. \(12 \text{ cm}\)
C. \(13 \text{ cm}\)
D. \(14 \text{ cm}\)
Solution: D
Let the width of rectangle is \(x\), then the length is \(2x - 2\)
\begin{equation*} \begin{split} \text{Area of the rectangle}& = \text{length} \times \text{width}\\\\ 112 & = (2x - 2)(x)\\\\ 112& = 2x^2 - 2x\:\:\:\:\:\color{blue}\text{(divide both sides by 2)}\\\\ 56& = x^2 - x\\\\ x^2 - x - 56& = 0 \:\:\:\:\:\color{blue}\text{(factorize)}\\\\ (x - 8)(x + 7)& = 0\\\\ x- 8 = 0 &\rightarrow x = 8\:\:\:\color{blue}\text{true}\\\\ x + 7 = 0 &\rightarrow \xcancel{x = -7}\\\\\\ \text{So, the width} & = 8 \text{ cm}\\\\ \text{The length}& = 2(8) - 2 = 14 \text{ cm} \end{split} \end{equation*}
The coefficient of the \(3^{rd}\) term of \((2a - 5b)^5\) is ...
(A) \(1000\)
(B) \(2000\)
(C) \(2500\)
(D) \(3000\)
The coefficient of the \(3^{rd}\) term of \((2a - 5b)^5\) is ...
(A) \(1000\)
(B) \(2000\)
(C) \(2500\)
(D) \(3000\)
Solution: B
\((2a - 5b)^5 = (2a)^5 + 5 \:.\: (2a)^4 \:.\: (-5b) + 10 \:.\: (2a)^3 \:.\: (-5b)^2 + 10 \:.\: (2a)^2 \:.\: (-5b)^3 + 5 \:.\: (2a) \:.\: (-5b)^4 + (-5b)^5\)
The third term is \(10 \:.\: (2a)^3 \:.\: (-5b)^2\)
\begin{equation*} \begin{split} & 10 \:.\: 8 a^8 \:.\: 25b^2\\\\ & 2000 \:.\: a^8 \:.\: b^2\\\\\\ \end{split} \end{equation*}