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Polynomials

Polynomials

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The Basics of Polynomials

Polynomials are mathematical expressions that contain a variable raised to different exponents, with a coefficient accompanying each term. They are typically written in the standard form of:

axn + bxn-1 + ... + rx + s

The terms are arranged in descending order of exponents, with the highest power being the degree of the polynomial. It is not necessary for all terms to be present, and any missing terms have a coefficient of zero.

For example, the polynomial 3x2 + 5x - 2 can also be written as 3x2 + 5x + 0 without changing its value.

Evaluating Polynomials

Evaluating a polynomial simply requires substituting the given value for the variable and solving for the result.

For instance, if we have the polynomial 2x3 + 5x2 - 3x + 4 and are asked to evaluate it at x = 2, it would be:

2(2)3 + 5(2)2 - 3(2) + 4 = 16 + 20 - 6 + 4 = 34

Adding and Subtracting Polynomials

When adding or subtracting polynomials, it is important to group similar terms and perform the operations correctly. For example:

  • 2x2 + 5x + 3
  • + 3x2 - 2x - 5
  • = (2x2 + 3x2) + (5x - 2x) + (3 - 5)
  • = 5x2 + 3x - 2

Factoring Polynomials

Factoring involves rewriting a polynomial as the product of simpler terms. The approach varies based on the degree of the polynomial and the coefficient of the first term.

For polynomials with a degree higher than two and a non-one coefficient on the first term, grouping and factoring can be used to simplify the expression.

For instance, to factor 5x2 + 16x + 3, we can rearrange the terms and group them as follows:

(5x2 + 15x) + (x + 3) = 5x(x + 3) + (x + 3) = (5x + 1)(x + 3)

If the first term has a coefficient of one, we can use the ac method to factor the polynomial. This involves finding two numbers that multiply to equal the product of the first and last terms and add or subtract to equal the coefficient of the middle term.

For example, to factor x2 + 6x + 8, we can follow these steps:

x2 + 6x + 8 = (x + 4)(x + 2) = 1x2 + (3 + 1)x + 2x2 + 4 = x(x + 4) + 2(x + 4) = (x + 2)(x + 4)

Dividing Polynomials

Dividing polynomials follows a similar process to long division. The dividend is divided by the divisor, and any remaining terms become the remainder.

Prior to division, we must ensure that the polynomial is in standard form with all missing terms having a coefficient of zero.

For example, to divide x2 + 4x + 6 by x + 2, we would follow these steps:

  • Arrange the terms in descending order: x2 + 4x + 6
  • Determine the first term of the quotient by dividing the first term of the dividend by the first term of the divisor: x2/x = x
  • Multiply the quotient by the divisor and subtract the result from the corresponding terms in the dividend: x(x + 2) = x2 + 2x, so (x2 + 2x) - (x2 + 4x) = -2x
  • Bring down the next term from the dividend and repeat the process until the remainder is zero: -2x + 6 - (-2x + 4) = 2

The Factor Theorem

The factor theorem is a useful tool when factoring polynomials. It states that if a value is substituted for the variable and the result is zero, then that value is a root of the polynomial, and (x - p) is a factor.

For example, to show that x - 3 is a factor of x2 - 9, we can substitute 3 for x and get (3)2 - 9 = 0, confirming that 3 is a root and x - 3 is a factor of x2 - 9.

We can also use the factor theorem to quickly divide a polynomial by a factor.

Polynomials are complex mathematical expressions that involve variables raised to different powers and multiplied by coefficients. To effectively work with polynomials, it is important to understand their key characteristics and the specific steps for different operations.

Key Features of Polynomials

  • A polynomial's standard form is written in descending order of exponents, starting from the largest.
  • Substituting values for the variables is necessary to evaluate polynomials.
  • When adding or subtracting polynomials, grouping like terms together and following the correct operations is crucial.
  • Factoring polynomials requires simplifying and rewriting them as the product of simpler terms.
  • Multiplying polynomials involves expanding and combining like terms.

How to Divide Polynomials

To divide polynomials, the first term of the dividend is divided by the first term of the divisor, resulting in the first term of the quotient. This term is then multiplied by each term in the divisor and subtracted from the dividend. The process is repeated until the degree of the final expression is lower than that of the divisor.

How to Factor Polynomials

The process of factoring polynomials differs depending on the degree and coefficient of the highest term. Generally, it involves simplifying and rewriting the polynomial as a product of two or more terms.

Solving Polynomials

When solving polynomials, the specific task and the degree of the highest term determine the approach. Simplifying and factoring the polynomial is typically necessary to reach a solution.

How to Multiply Polynomials

Multiplying polynomials requires expanding and simplifying the brackets, and then combining like terms to obtain the final solution.

In conclusion, understanding the key features and following the correct steps for operations such as division, factoring, solving, and multiplication is crucial for effectively working with polynomials. By doing so, accurate results can be achieved and complex mathematical expressions can be successfully manipulated.

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