# Algebra foundations questions and answers

Recent questions in Algebra foundations
Algebra foundations

### To factorize: $$\displaystyle{5}{x}^{{{3}}}{y}-{15}{x}^{{{2}}}{y}+{10}{x}{y}$$

Algebra foundations

### Find the sloope if it exist, $$\displaystyle{4}{x}-{y}={8}$$

Algebra foundations

### Foundations/Comprehension 1.The following limits are of the form 0/0. Using algebraic methods, simplify the expressions, then compute the limits (5 points each). a)$$\displaystyle\lim_{{{x}\rightarrow{10}}}{\frac{{{3}{x}^{{{2}}}-{23}{x}-{70}}}{{{11}{x}^{{{2}}}-{1100}}}}$$ b)$$\displaystyle\lim_{{{h}\rightarrow{0}}}{\frac{{{5}{\left({8}+{h}\right)}^{{{2}}}-{320}}}{{h}}}$$

Algebra foundations

### The value of the logarithmic expression $$\displaystyle{{\log}_{{{2}}}{16}}$$

Algebra foundations

### To write: An algebraic expression for given phrase. The product of -6 and the sum of a number and 15.

Algebra foundations

### Translate the given phrase into algebraic expression. A number subtracted from $$\displaystyle-{\frac{{{3}}}{{{8}}}}$$

Algebra foundations

### The multiplication of the expression $$\displaystyle{\left({8}{x}-{3}\right)}{\left({2}{x}-{4}\right)}$$

Algebra foundations

### To write: an algebraic expression for given phrase. The quotient of -20 and a number, increased by three.

Algebra foundations

### Let $$R=\left\{\begin{bmatrix}a & b \\0 & a \end{bmatrix}; a,b\in Q\right\}$$. One can prove that R with the usual matrix addition and multiplication is a ring. Consider $$J =\left\{\begin{bmatrix}0 & b \\0 & 0 \end{bmatrix};b\in Q\right\}$$ be a subset of the ring R. (a) Prove that the subset J is an ideal of the ring R; (b) Prove that the quotient ring $$\displaystyle\frac{R}{{J}}$$ is isomorphic to Q.

Algebra foundations

### To translate given phrase into an algebraic expression using $$\displaystyle{x}$$ to represent the number. The given phrase is divide a number by $$\displaystyle-{6}{\frac{{{1}}}{{{11}}}}$$.

Algebra foundations

### Simplify, $$5\left\{\left[4(q-4)+18\right]-\left[2(5q-2)+3\right]\right\}$$

Algebra foundations

### To find: The nuclear waste left after 10 century. The nuclear waste from an atomic energy plant decays at arate of $$\displaystyle{3}\%$$ each century. If 150 pounds of nuclear waste is disposed initially.

Algebra foundations

### Solve the following equation. $$\displaystyle{3}-{4}{\left({x}+{1}\right)}-{3}={3}+{2}{\left({4}-{2}{x}\right)}-{16}$$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is $$\displaystyle{x}={B}\otimes$$. (Simplify your answer.) B. Every real number is a solution. C. There is no solution.

Algebra foundations

### Find the slope and y-intercept of the following equation $$\displaystyle{3}{y}-{5}={2}{\left({x}-{2}\right)}$$ Slope= ? y-Intercept=? Q)Find the slope and y-intercept of the following equation $$\displaystyle{15}{x}+{3}{y}={2.4}$$ Slope= ? y-Intercept=? Use algebra to solve the following systems of equations A)$$\displaystyle{12}{y}={5}{x}+{16}$$ $$\displaystyle{6}{x}+{10}{y}-{54}={0}$$ B)$$\displaystyle{\frac{{{9}{x}}}{{{5}}}}+{\frac{{{5}{y}}}{{{4}}}}={\frac{{{47}}}{{{10}}}}$$ $$\displaystyle{\frac{{{2}{x}}}{{{9}}}}+{\frac{{{3}{y}}}{{{8}}}}={\frac{{{5}}}{{{36}}}}$$ C)$$\displaystyle{0.2}{a}+{0.3}{b}={0}$$ $$\displaystyle{0.7}{a}+{0.2}{b}={259}$$ 1.General equation for slope is $$\displaystyle{y}={m}{x}+{b}$$. Where m is the slope and b is the y-intercept. $$\displaystyle{3}{y}-{5}={2}{\left({x}-{2}\right)}$$ $$\displaystyle{3}{y}-{5}={2}{x}-{4}$$ Simplifying, $$\displaystyle{3}{y}={2}{x}+{1}$$. $$\displaystyle{y}=\frac{{2}}{{3}}\cdot+\frac{{1}}{{3}}$$. Comparing the above equation with the generalised equation, we get, 2/3 as slope and 1 as the y-intercept. $$\displaystyle{2.15}{x}+{3}{y}={2.4}$$ $$\displaystyle{3}{y}=-{1.5}\cdot{x}+{2.4}$$ $$\displaystyle{y}=-{0.5}\cdot{x}+{0.8}$$ Comparing the above equation with the generalised equation we get slope as -0.5 and y-intercept as 0.8

Algebra foundations

### Simplify. $$8\left\{\left[4(p-4)+19\right]-\left[2(3p-2)+5\right]\right\}$$

Algebra foundations

### Let $$A=\begin{bmatrix}4 & 0 &5 \\-1 & 3 & 2 \end{bmatrix}$$, $$B=\begin{bmatrix}1 & 1 &1 \\3 & 5 & 7 \end{bmatrix}$$, $$C=\begin{bmatrix}2 & -3 \\0 & 1 \end{bmatrix}$$ Find: $$\displaystyle{3}{A}-{B}$$, and $$\displaystyle{C}\times{B}+{A}$$

Algebra foundations

### For a,b $$\displaystyle\in$$Z, let B(a,b) $$\displaystyle\in$$M(2,Z) be defined by $$\displaystyle{B}{\left({a},{b}\right)}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{a}&{3}{b}\backslash{b}&{a}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$. Let $$\displaystyle{S}=\le{f}{t}{\left\lbrace{B}{\left({a},{b}\right)};{a},{b}\in{Z}{r}{i}{g}{h}{t}\right\rbrace}\subseteq$$ M(2,Z). Show that $$\displaystyle{S}\stackrel{\sim}{=}{Z}{\left[\sqrt{{{3}}}\right]}=\le{f}{t}{\left\lbrace{a}+{b}\sqrt{{{3}}};{a},{b}\in{Z}{r}{i}{g}{h}{t}\right\rbrace}$$.

Algebra foundations

### Complete the table by the use of indicated operations. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Monomials}&\text{Add}&\text{Subtract}&\text{Multiply}&\text{Divide}\backslash{h}{l}\in{e}{6}{x},\ {3}{x}&{6}{x}+{3}{x}={9}{x}&{6}{x}-{3}{x}={3}{x}&{6}{x}\cdot{3}{x}={18}{x}^{{{2}}}&{\frac{{{6}{x}}}{{{3}{x}}}}={2}\backslash{h}{l}\in{e}-{12}{x}^{{{2}}},\ {2}{x}&-{12}{x}^{{{2}}}-{2}{x},\ \text{can't be simplified}&-{12}{x}^{{{2}}}\cdot{2}{x}=-{24}{x}^{{{3}}}&{\frac{{-{12}{x}^{{{2}}}}}{{{2}{x}}}}=-{6}{x}\backslash{h}{l}\in{e}{5}{a},\ {15}{a}&\backslash{h}{l}\in{e}{4}{y}^{{{2}}},\ {4}{y}^{{{3}}}&\backslash{h}{l}\in{e}-{3}{y}^{{{5}}},\ {9}{y}^{{{4}}}&\backslash{h}{l}\in{e}-{14}{x}^{{{2}}},\ {2}{x}^{{{2}}}&\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$

Algebra foundations