POLYNOMIAL

 POLYNOMIALS

Definition of a Polynomial

A polynomial is an algebraic expression consisting of variables, coefficients, and exponents that involve only non-negative integers. It can be written in the general form:

$P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$

where:

• $a_n, a_{n-1}, ..., a_0$ are constants (coefficients)
• $x$ is the variable
• $n$ is a non-negative integer (degree of the polynomial)

Examples:

1. $3x^4 - 2x^3 + 5x - 7$ (Polynomial of degree 4)
2. $x^2 - 4x + 4$ (Polynomial of degree 2, quadratic polynomial)
3. $7x^5 + 3x^2 - x + 8$ (Polynomial of degree 5)

Addition and Subtraction of Polynomials

To add or subtract polynomials, we combine like terms (terms with the same variable and exponent).

Example 1:

Simplify $(3x^2 + 4x - 5) + (2x^2 - 6x + 3)$

Solution: $(3x^2 + 4x - 5) + (2x^2 - 6x + 3)$ $= (3x^2 + 2x^2) + (4x - 6x) + (-5 + 3)$ $= 5x^2 - 2x - 2$

Example 2:

Simplify $(5x^3 - 3x^2 + 2x) - (2x^3 - x^2 + 4)$

Solution: $(5x^3 - 3x^2 + 2x) - (2x^3 - x^2 + 4)$ $= 5x^3 - 3x^2 + 2x - 2x^3 + x^2 - 4$ $= (5x^3 - 2x^3) + (-3x^2 + x^2) + 2x - 4$ $= 3x^3 - 2x^2 + 2x - 4$

Multiplication of Polynomials

Polynomials are multiplied using the distributive property or the FOIL method (for binomials).

Example 3:

Multiply $(x + 2)(x - 3)$

Solution: $(x + 2)(x - 3)$ Using the distributive property: $x(x - 3) + 2(x - 3)$ $= x^2 - 3x + 2x - 6$ $= x^2 - x - 6$

Division of Polynomials

Polynomials can be divided using long division or synthetic division.

Example 4:

Divide $(2x^3 + 3x^2 - 5x + 6)$ by $(x - 2)$

Solution: Using long division:

1. Divide $2x^3$ by $x$: $2x^2$
2. Multiply: $2x^2(x - 2) = 2x^3 - 4x^2$
3. Subtract: $(3x^2 - 5x + 6) - (2x^3 - 4x^2)$ $= 7x^2 - 5x + 6$
4. Repeat the process for $7x^2$, $-5x$, and $6$ Final quotient: $2x^2 + 7x + 9$, remainder $24$

Factor Theorem and Remainder Theorem

Factor Theorem

The factor theorem states that if $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$.

Example 5:

Show that $(x - 2)$ is a factor of $P(x) = x^3 - 3x^2 + x + 5$

Solution: Substituting $x = 2$: $P(2) = 2^3 - 3(2^2) + 2 + 5$ $= 8 - 12 + 2 + 5 = 3$ Since $P(2) \neq 0$, $(x - 2)$ is not a factor.

Remainder Theorem

The remainder theorem states that if a polynomial $P(x)$ is divided by $(x - a)$, the remainder is $P(a)$.

Example 6:

Find the remainder when $P(x) = x^3 - 4x^2 + 5x - 2$ is divided by $x - 3$

Solution: Substituting $x = 3$: $P(3) = 3^3 - 4(3^2) + 5(3) - 2$ $= 27 - 36 + 15 - 2$ $= 4$ Thus, the remainder is 4.

Conclusion

Polynomials form the foundation of algebraic expressions. Understanding operations such as addition, subtraction, multiplication, and division is crucial for solving polynomial equations. The factor and remainder theorems are powerful tools in determining factors and simplifying polynomial division. Additionally, these theorems provide a structured approach to identifying the roots and behavior of polynomial functions in algebraic analysis.

Polynomial JAMB Quiz