Polynomial is an important concept in algebra, and it is a type of mathematical expression that contains one or more variables and coefficients. In this guide, we will explore the idea of polynomial numbers and the four basic arithmetic operations that ca...

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Polynomial is an important concept in algebra, and it is a type of mathematical expression that contains one or more variables and coefficients. In this guide, we will explore the idea of polynomial numbers and the four basic arithmetic operations that can be performed on them: addition, subtraction, multiplication, and division.

A polynomial is an algebraic expression that consists of one or more terms, where each term is a product of a coefficient and one or more variables raised to a power. For example, the polynomial \(3x^2 +2x-5\) contains three terms: \(3x^2\), 2x, and -5. The coefficient of each term is 3, 2, and -5 respectively. The variable in this case is x, and it is raised to the power of 2 in the first term and to the power of 1 in the second term.

Polynomials can have one or more variables, and each variable can have a different power. For instance, the polynomial \(4x^2+3xy^2-2x^3z\) contains three terms with three variables: x, y, and z. The coefficient of each term is 4, 3, and -2, respectively. The variable x is raised to the power of 2 in the first term and to the power of 3 in the third term. Similarly, y is raised to the power of 1 in the first term and to the power of 2 in the second term, and z is raised to the power of 1 in the third term.

**a)** **Addition of polynomial numbers:**

The addition of polynomial numbers is performed by adding the coefficients of the same variables and their respective powers. For example, consider the addition of the polynomials \(3x^2+2x-5\) and \(2x^2-3x+1\). To add these polynomials, we simply add the coefficients of the same terms. Thus, the sum of these polynomials is (3+2)\(x^2\) + (2-3)x + (-5+1) = \(5x^2-x-4\).

**b)** **Subtraction of polynomial numbers:**

The subtraction of polynomial numbers is performed by subtracting the coefficients of the same variables and their respective powers. For example, consider the subtraction of the polynomials \(3x^2+2x-5\) and \(2x^2-3x+1\). To subtract these polynomials, we simply subtract the coefficients of the same terms. Thus, the difference of these polynomials is (3-2)\(x^2\) + (2+3)\(x\) + (-5-1) = \(x^2+5x-6\).

**c)** **Multiplication of polynomial numbers:**

The multiplication of polynomial numbers is performed by multiplying each term of one polynomial with each term of the other polynomial and adding the resulting terms. For example, consider the multiplication of the polynomials (x+2)(x-3). To multiply these polynomials, we multiply each term of the first polynomial (x and 2) with each term of the second polynomial (x and -3) and then add the resulting terms. Thus, (x+2)(x-3) = \(x^2-3x+2x-6 = x^2-x-6\).

**d) Division of polynomial numbers:**

The division of polynomial numbers is performed by long division, similar to the long division of whole numbers. For example, consider the division of the polynomial \(x^3-2x\) by x-1. To perform the division, we first divide the highest degree term of the dividend by the highest degree term of the divisor. This gives us \(x^2\) as the first term of the quotient. We then multiply the divisor by \(x^2\) and subtract the result from the dividend. This gives us a new dividend of \(x^2-2x\), which we repeat the process for. We divide the highest degree term of the new dividend, which is \(x^2\), by the highest degree term of the divisor, which is x. This gives us x as the second term of the quotient. We then multiply the divisor by x and subtract the result from the new dividend, which gives us a remainder of -x. The final quotient is \(x^2+x\), with a remainder of -x. Therefore, \(x^3-2x\) divided by x-1 is equal to \(x^2+x\) with a remainder of -x.

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