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Evaluating Polynomials: Adding, Subtracting, Multiplying, and Dividing

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Evaluating Polynomials: Adding, Subtracting, Multiplying, and Dividing

Polynomials are expressions with multiple terms, each consisting of a coefficient and a variable raised to a non-negative integer power. Evaluating polynomials involves substituting values for variables and simplifying the expression. This process is essential in various mathematical fields, including algebra, calculus, and physics.

Adding and Subtracting Polynomials

Adding and subtracting polynomials is straightforward. You simply combine like terms, which are terms with the same variable and exponent. For example:

(3x2 + 2x - 1) + (5x2 - 4x + 2) = 8x2 - 2x + 1

To subtract polynomials, distribute the negative sign and then combine like terms:

(3x2 + 2x - 1) - (5x2 - 4x + 2) = 3x2 + 2x - 1 - 5x2 + 4x - 2 = -2x2 + 6x - 3

Multiplying Polynomials

Multiplying polynomials requires using the distributive property. You multiply each term in the first polynomial by each term in the second polynomial and then combine like terms. For example:

(x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x2 - 3x + 2x - 6 = x2 - x - 6

Dividing Polynomials

Dividing polynomials is more complex than adding, subtracting, or multiplying. There are several methods for polynomial division, including long division and synthetic division. Long division is a more general method, while synthetic division works only for linear divisors.

Long Division

Long division for polynomials is similar to long division for numbers. Here's an example:

Divide (x3 + 2x2 - 5x + 1) by (x - 2):

```
x2 + 4x + 3
x - 2 | x3 + 2x2 - 5x + 1
-(x3 - 2x2)
----------------
4x2 - 5x
-(4x2 - 8x)
----------------
3x + 1
-(3x - 6)
----------------
7
```

Therefore, (x3 + 2x2 - 5x + 1) divided by (x - 2) is x2 + 4x + 3 with a remainder of 7.

Synthetic Division

Synthetic division is a shortcut for dividing polynomials by linear divisors. It involves using only the coefficients of the polynomials and a simple set of steps. Here's an example:

Divide (x3 + 2x2 - 5x + 1) by (x - 2):

```
2 | 1 2 -5 1
| 2 8 6
----------------
1 4 3 7
```

The result of the division is 1x2 + 4x + 3 with a remainder of 7.

Conclusion

Evaluating polynomials is a fundamental skill in mathematics. By understanding the techniques for adding, subtracting, multiplying, and dividing polynomials, you can solve a wide range of problems and develop a deeper understanding of algebraic concepts.

For further learning on polynomial division, you can refer to these resources:

Polynomials