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The degree of a polynomial is a fundamental concept that signifies the highest power of the variable in the polynomial. For the given polynomial

Pon

P_{6}(x)=x^{6}+4 x^{5}-8 x^{4}-34 x^{3}+57 x^{2}+130 x-150,

the highest exponent is 6. Therefore, the degree of this polynomial is 6.

Understanding Degrees: The degree can tell you important things about the polynomial, like how it behaves for very large or small values of x.

In this polynomial, because it is of the 6th degree, the graph of the polynomial will have a maximum of 6 - 1 = 5 turning points (places where the graph changes direction). It will also have up to 6 roots (places where the graph crosses the x-axis).

The substitution method is a straightforward technique to evaluate polynomials at specific values of x. By replacing the variable x with a given number, you can directly calculate the value of the polynomial.

How to Use Substitution: Follow these simple steps:

• Take the given polynomial: P_{6}(x)=x^{6}+4 x^{5}-8 x^{4}-34 x^{3}+57 x^{2}+130 x-150
• Choose a value for x. For example, let x = 1.
• Replace every instance of x in the polynomial with 1.
• Perform the arithmetic to solve for the value at that specific x.

Substituting x = 1 into the polynomial: \[ P_{6}(1) = 1^6 + 4 \times 1^5 - 8 \times 1^4 - 34 \times 1^3 + 57 \times 1^2 + 130 \times 1 - 150 \] Performing the calculations in sequence: \[ P_{6}(1) = 1 + 4 - 8 - 34 + 57 + 130 - 150 \] Finally: \[ P_{6}(1) = 0 \]
This tells you that when x = 1, the value of the polynomial is 0.

Evaluating polynomials means finding the polynomial's value at a specific point. This is typically done using the substitution method.

To evaluate a polynomial:

• Select a value for x.
• Substitute this value into the polynomial.
• Carry out the arithmetic operations in sequence, following the order of operations (brackets, exponents, multiplication and division, addition and subtraction).

Example: Use the previous example: \[ P_{6}(x)=x^{6}+4 x^{5}-8 x^{4}-34 x^{3}+57 x^{2}+130 x-150 \] If you need to evaluate it for x = 2:

First, substitute 2 in place of x:

\[ P_{6}(2) = 2^6 + 4 \times 2^5 - 8 \times 2^4 - 34 \times 2^3 + 57 \times 2^2 + 130 \times 2 - 150 \]

Next, perform the operations in sequence:

\[ P_{6}(2) = 64 + 128 - 128 - 272 + 228 + 260 - 150 \] Finally: \[ P_{6}(2) = 130 \]

Knowing how to evaluate polynomials can be incredibly useful in solving many mathematical problems and real-world applications.