Algebra 1 is a branch of mathematics in which we represent the components of a specified set or the numbers by use of symbols, alphabetical letters to express general relationship between the components of set. Algebra is a way of simplifying Math through the use of variables, formulas or equations. Algebra 1 is a broad topic which covers several topics like solving rational equations, algebraic equations, linear and quadratic equations etc.
This topic contains Rational Numbers on number line, addition $(+)$, subtraction $(-)$ multiplication $(\times)$, division of rational numbers, square root $(\sqrt{x})$ and Cube root $(x^{\frac{1}{3}})$ of Real Numbers.
This topic contains solution of various linear equations by using property of addition $(2x + 3x = 5x)$, subtraction $(5x - 2x = 3x)$, multiplication $(7x \times 8x = 56 x^2)$ and division ($\frac{12x}{4x}$ = $3$) with variable on each side. Also, the properties of ratio $(a : b)$ and proportion $(a : b :: c : d)$, percentage $( \%)$ come under this branch.Factoring: This topic contains greatest common factor, factorizing of trinomials $(ax^{2} + bx + c)$ by splitting middle term or by the method of perfect square. Statistics: Probability: |

**Polynomials:**

In simple words, polynomial means, 'many terms'. They are the expression which include monomials, binomials, trinomial and also the higher order of polynomials. There are different types of polynomials and they are classified on the bases of number of the terms and the degree. Given below are the different types of Polynomials:

**1) Monic Polynomial:**

**2) Quadratic Polynomial:**

The general expression of quadratic polynomial is $ax^{2} + bx + c, a \neq 0$

**3) Cubic Polynomial:**

**4) Prime Polynomial:**

**Ex:**$x^{2} + x + 1$

**Radical Expression**

An expression containing a square root is called as radical expression. These expressions can be easily solved by performing squaring operation.

**Rational Expression**

An expression where the numerator and denominator or both of them are polynomials are called as rational expressions. By simplifying the rational expression, we can reduce the expression into the lowest form.

**Analyzing Linear Equations**

A linear equation is an equation of a line, which contains two different variables and is easily transferred into graphical representation. The general form of linear equation is $ax + by + c$ = $0$, where $x$ and $y$ are variables, $a$ and $b$ are co-efficients of $x$ and $y$ and $c$ is constant.

Solving Linear Inequalities

Solving Linear Inequalities

Linear inequality is the comparison of two values. We can add, subtract, multiply and divide the inequalities to solve the linear inequalities. The concept of solving the linear inequalities is the same as solving the linear equation. Linear inequality has its own properties and they are as given below:

**1) Transitive Property**

**2) Addition Property:**

**3) Subtraction Property:**

**4) Multiplication Property:**

**•**If $a < b$ and $c$ is positive $c \times a < c \times b$

**•**If $a < b$ and $c$ is negative - $c \times a > - c \times b$

**Quadratic Equation**

A polynomial with a second degree is called as quadratic equation. When the discriminant is zero, then such quadratic equation is called as perfect square, i.e. $b^{2} - 4ac$ = $0$. When a discriminant is greater than $1$ then the roots of the equation are real and if it is lesser than $1$, at that point the roots of the equation are imaginary. The general form of quadratic equation is $ax^{2} + bx + c$ = $0$. And, the formula of quadratic equation is as follows:

**Simultaneous Equation**

**1) Substitution Method:**

Here, we first solve one of the equation for one of the unknowns and then, substitute the answer into the other equation.

**2) Elimination Method:**

In this method, first make the co-efficient of one of the variables to the same value in both the equations. Then, add or subtract an equation from the other to form a new equation that contains one variable.

### Solved Examples

**Question 1:**Subtract 2x

^{3 }+ x + 1 from 16x

^{3 }+ 4x + 1

**Solution:**

**Given:**16x

^{3 }+ 4x + 1 and 2x

^{3 }+ x + 1

Here first, let us find the opposite of the subtracted term

That is, 2x

^{3 }+ x + 1 = - 2x

^{3 }- x - 1

By subtracting the two equations, we get

16x

^{3 }+ 4x + 1 - 2x

^{3 }- x - 1

= (16x

^{3 }- 2x

^{3}) + (4x - x) + 1 - 1

= 14x

^{3 }+ 3x

**Question 2:**Solve the radical expression $\sqrt{5y - 7}$ = y - 1

**Solution:**

**Given:**$\sqrt{5y - 7}$ = y - 1

First, we need to square both the sides

$\left [ \sqrt{5y - 7} \right ]^2$ = $(y - 1)^2$

By simplifying the above equation, we get

5y - 7 = y

^{2 }- 2y + 1

After converting the above equation into factor form, we get

y

^{2 }- 7y + 8 = 0

We got the above equation in the quadratic equation. Let us solve this equation using the quadratic equation

x = $\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

Here, a = 1, b = -7, c = 8

$\therefore$ x

_{ }= $\frac{7\pm \sqrt{7^{2}-4.1.8}}{2}$

x = $\frac{7\pm \sqrt{49 - 32}}{2}$

x

_{ }= $\frac{7\pm \sqrt{17}}{2}$

x

_{1}= $\frac{7+ \sqrt{17}}{2}$ and x

_{2 }= $\frac{7- \sqrt{17}}{2}$

**Question 3:**Solve the linear inequalities 4x + 28 > 0

**Solution:**

**Given:**4x + 28 > 0

First, subtract 28 on both the sides.

4x + 28 - 28 > 0 - 28

4x + 0 > 0 - 28

4x > -28

Now, divide 4 on both sides,

$\frac{4x}{4}$ > $\frac{-28}{4}$

x > -7

**Question 4:**Solve the following simultaneous equations using the elimination method

6x + 7y = 12

4x - 7y = 8

**Solution:**

**Given:**

6x + 7y = 12 --> 1

4x - 7y = 8 --> 2

Since the co-efficient of one of the variables is numerically equal and also opposite signs, we can add 2 equations for eliminating y variable.

By adding the like term, we get

(6x + 4x) + (7y - 7y) = 12 + 8

That is, 10x = 20

Now, let us divide by 10 both the sides

$\frac{10x}{10}$ = $\frac{20}{10}$

$\therefore$ x = 2

Now, substitute the value of x in the first equation

6(2) + 7y = 12

12 + 7y = 12

12 + 7y -12 = 12 - 12

7y = $\frac{0}{7}$

y = 0

=> x = 2, y = 0