1. Definition of algebraic expression:
An algebraic expression is a mathematical term that combines constants and variables using operations like addition, subtraction, multiplication, division.
Constants:
A constant is a fixed numerical value.
Variable:
A symbol which takes on various numerical values is knowns as variable or literal.
Example:
\(a + b\) , \(x^2 + x + 5\)
2. Definition of Polynomial:
- An expression of the form \(p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where \(n\) is an non negative integer and \(a_0, a_1, a_2, ..., a_n\) are constants and \(a_n \neq 0\) is called a polynomial in \(x\) of degree \(n\).
- An algebraic expression involving one variable and its power are called one - variable polynomials, univariate polynomials or when the context is clear, simply polynomials.
Example:
\(p(x) = x^7 - 3x^6 + x^3 +x + 9\)
The above example is an univariate polynomial.
3. Degree of a polynomial:
If \(p(x)\) is a polynomial in \(x\), then the heightest power of \(x\) in \(p(x)\) is called degree of a polynomial \(p(x)\).
Example:
Degree of the given polynomial \(p(x) = x^7 - 3x^6 + x^3 +x + 9\) is \(7\).
4. Terms and Coefficients of Polynomials
In a polynomial, \(\text{terms}\) are the individual parts separated by plus (+) or minus (-) signs, which can contain numbers, variables, or their products.
\(\text{Coefficients}\) refer to the numerical values that are multiplied by the variable.
Example:
\(3x+5\)
Here the terms are \(3x\) and \(5\), and the coefficient of \(x\) is \(3\).
5. Types of polynomials using degree:
Linear polynomial:
A polynomial with degree \(\text{one}\) is called linear polynomial.
Example:
\(3x+5\)
Quadratic polynomial:
A polynomial with degree \(\text{two}\) is called quadratic polynomial.
Example:
\(3x^2 + 2x +5\)
Cubic polynomial:
A polynomial with degree \(\text{three}\) is called cubic polynomial.
Example:
\(3x^3 + 5x^2 + 4x + 9\)
6. Types of polynomials using number of terms:
Monomials:
A polynomial having only \(\text{one}\) term is called monomials.
Example:
\(3x\), \(5a\), \(x^{100}\)
Binomials:
A polynomials having two terms is called binomials.
Example:
\(2x + 4\), \(x^2 - 1\)
Trinomials:
A polynomials having \(\text{three}\) terms is called trinomials.
Example:
\(2x^2 + 3x + 4\), \(x^3 + 5x^2 + 9x\)
7. Linear polynomial
A polynomial is considered to be a linear polynomial if it has its degree as \(1\).
Example:
\(x+1\), \(4x+2\)
8. Linear Patterns:
A linear pattern is a pattern in which numbers increase or decrease by the same constant value for each time.
A linear pattern is denoted in the form of \(y = ax + b\).
Example:
Observe the table to find the value of y.
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) |
| \(y\) | \(5\) | \(8\) | \(11\) | \(14\) | \(17\) | \(?\) |
When \(x = 1\), \(y = 5 = 3 \times 1 + 2\)
When \(x = 2\), \(y = 8 = 3 \times 2 + 2\)
When \(x = 3\), \(y = 11 = 3 \times 3 + 2\)
When \(x = 4\), \(y = 14 = 3 \times 4 + 2\)
When \(x =5\), \(y = 17 = 3 \times 5 + 2\)
In above term each term has a difference of \(3\)
Using the above pattern we get,
When \(x = 6\), \(y = 3 \times 6 + 2 = 20\)
The linear pattern formed using the above table is,
\(y = 3x + 2\)
9. Linear equations:
When a linear polynomial is equal to \(0\) is called linear equations. And the value which statisfies that equation is called its root or zero.
Example:
\(ax + by + c = 0\) where \(a \neq 0\), \(b \neq 0\)
10. Linear Growth (Growing Steadily)
Linear Growth happens when something increases by the exact same amount every single time.
For example hiring an auto-rickshaw or a taxi,
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You pay a fixed starting charge of \(₹80\) just to sit in the taxi (even if you travel \(0\) km).
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For every \(1\) km you travel, the price goes up by a fixed amount of \(₹50\).
This gives us a simple rule (or formula):
\(\text{Total Cost } C(d) = 80 + 50d\)
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If you travel \(1\) km Cost \(= 80 + 50(1) = \text{₹}130\)
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If you travel \(2\) km: Cost \(= 80 + 50(2) = \text{₹}180\)
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Notice how the cost increases by exactly \(₹50\) every time? That steady, fixed increase is Linear Growth.
11. Linear Decay (Shrinking Steadily)
Linear Decay is just the opposite of linear growth. It happens when something decreases by the exact same amount every single time.
For example, let us consider a water tank drying up during the hot summer months
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At the start \(t = 0\), the water level is \(2\) meters high.
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Every month, the hot sun causes the water level to drop by exactly \(0.25\) meters.
The rule for this situation is:
\(\text{Water Height } h(t) = 2 - 0.25t\)
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After \(1\) month: Height \(= 2 - 0.25(1) = 1.75 \text{ m}\)
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After \(2\) months: Height \(= 2 - 0.25(2) = 1.50 \text{ m}\)
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Because the water level goes down by the exact same \(0.25\) meters every month, this steady decrease is Linear Decay.