Introduction to Linear PolynomialsClass 9 Mathematics Important Points

A univariate polynomial is an algebraic expression with only one variable, such as $x$ or $y$. An example is $5y^3 + y^2 + 2y - 1$, which is a polynomial in the variable $y$.

The degree of a polynomial is the highest power of the variable in the expression. For example, the polynomial $x^2 + 5x + 1$ has a degree of 2.

Polynomials are classified by their degree. A polynomial of degree 1 is linear (e.g., $3z+7$), degree 2 is quadratic (e.g., $x^2+5x+1$), and degree 3 is cubic (e.g., $5y^3+y^2-1$).

A constant number, like 8, is called a constant polynomial. Its degree is 0 because it can be written as $8x^0$.

A linear polynomial is a polynomial of degree 1. The general form is $ax+b$, where $a$ and $b$ are constants and the coefficient of $x$, which is $a$, is not zero.

To evaluate a polynomial, substitute a given value for the variable. For the polynomial $p(x) = 5x-3$, if $x=2$, the value is $p(2) = 5(2) - 3 = 7$.

A linear equation is formed when a linear polynomial is set equal to a constant value. For instance, equating the linear polynomial $2x+10$ to 64 gives the linear equation $2x+10=64$.

A linear pattern is a sequence of numbers where the difference between any two consecutive terms is constant. The rule for the $n^{ ext{th}}$ term of such a pattern is a linear polynomial, like $2n-1$.

Linear growth is when a quantity increases by a constant amount over equal intervals. Linear decay is when a quantity decreases by a constant amount. Both are modeled by linear relationships.

A linear relationship between two variables, $x$ and $y$, can be represented by the equation of a straight line: $y = ax+b$.

In the equation $y = ax+b$, the coefficient 'a' represents the slope of the line. A positive slope ($a>0$) indicates linear growth, while a negative slope ($a<0$) indicates linear decay.

In the equation $y = ax+b$, the constant 'b' is the y-intercept. This is the point $(0, b)$ where the line crosses the y-axis.

If a linear relationship $y=ax+b$ passes through two known points, you can find $a$ and $b$ by substituting the coordinates of both points into the equation to form and solve a pair of simultaneous equations.

Two lines are parallel if they have the same slope but different y-intercepts. For example, the lines $y = 3x+1$ and $y = 3x-1$ are parallel because both have a slope of $a=3$.

Any line with an equation of the form $y = ax$ (where the y-intercept $b=0$) will always pass through the origin, which is the point $(0,0)$.