↑ Return to 10 – Polynomials

# 10.1 – What is a Polynomial?

## Key Terms

• Degree – The value of the greatest exponent in a polynomial.
• Evaluate – To find the numerical value of an expression.
• Factor – A number or expression that can be multiplied by another number or expression to create a certain product.
• Factoring – Writing a number or expression as a product of two or more numbers or expressions.
• Monomial – A polynomial with only one term.
• Polynomial – An algebraic expression with one or more terms.
• None of the variables are in the denominator of a fraction, and any exponents are whole numbers.

## Review

Parts of Polynomials
• Degree: exponents, powers
• Coefficients: numbers that are before (and multiplied with) a variable.
• Ex. 3x: the coefficient is 3
• Ex. $-5a^{2}$: the coefficient is -5
• Constants: numbers without variables
• Nonnegative numbers: positive numbers or zero
• Terms: monomials separated by plus and minus signs
Integers
• Integers: Negative, Positive and Zero numbers
• $\frac{1}{2}$ is not an integer
• $\sqrt{x}=x^{\frac{1}{2}}$ is a root raised to a non-integer power
• -1 is a negative integer
• $\frac{x}{x-1}=x(x-1)^{-1}$ is a fraction with a negative exponent

## Notes

Monomials
• A monomial term always has the form $ax^{n}$, where a is any number and n is a nonnegative integer.
• The number a is called the coefficient of the term, and n is called the degree of the term.

Not Monomials
• Fractions with variables in the denominator
• Variables with negative exponents
• Radicals (√) with variables under the root

Polynomials
• One or more monomials separated by plus and minus signs
• Plus signs are positive numbers and minus signs are negative numbers
• Some polynomials have special names
• Monomials have 1 term (mono means 1)
• Binomials have 2 terms (bi means 2)
• Trinomials have 3 terms (tri means 3)
• Ex 1. $-4x^{3}-2x^{2}+6x^{1}-8x^{0}$ simplifies to $-4x^{3}-2x^{2}+6x-8$
• Notice that the x-term is to the 1st power, but you don’t have to write it.
• Notice that the constant term is to the zero power, but you don’t have to write it.
• Anything to the zero power equals 1; so, 8 times $x^{0}$ is the same as 8 times 1.
• 8 times 1 equals 8, so that’s why it is a constant term.
• Ex 2. $\frac{3}{5}x^{2}-2$ is a polynomial
• Remember, the coefficient can be a fraction, but the exponent cannot

Degrees and Exponents
• The greatest exponent in a polynomial is the degree of that polynomial.
• Ex. $-5x^{4}+2x^{3}-\frac{1}{3}x^{2}-18x+3$ has a degree of 4
• Ex. $9x^{18}+4x^{4}-55x--91$ has a degree of 18
• Ex. $3x+5$ has a degree of 1 since x is the same as x to the 1st power
• Ex. $99$ has a degree of zero because it is the same as $99x^{0}$ since $x^{0}=1$ and 99 times 1 = 99.

Descending Order
• You will need to rewrite polynomials in descending order.
• When a polynomial is written in descending order, its first term is called its leading term.
• The leading term has a leading coefficient.
• This is the order from the highest degree to the smallest degree (usually the constant term).
• How to put polynomials into descending order:

Evaluating Polynomials
• Steps to evaluate a polynomial
1. Substitute a value for the variable
2. Simplify
• Ex. Evaluate x = 2 in the polynomial: $3x^{3}-x^{2}+4x-6$
• $3(2)^{3}-(2)^{2}+4(2)-6$
• $3(8)-4+8-6$
• $24-4+8-6$
• $20+8-6$
• $28-6$
• Answer: 22

Factoring Polynomials
• Remember
• Factors are multiplied
• Terms are added

Closed Sets
• A set has closure under an operation if the performance of that operation on any members of the set always produces a member of the same set.
• Notice that division is NOT a closed set.

Examples
• Ex 1. The polynomial $6x^{2}+9x$ has factors of 3x and what?
• 3x times what will give you $6x^{2}$? Answer: 2x
• 3x times what will give you $9x$? Answer: 3
• Combined answer: $2x+3$
• Ex 2. Under which of the following operations are the polynomials $11x+3$ and $5y-8$ not closed?
• Answer: Division
• Ex 3. What is the coefficient of the term of degree 3 in this polynomial?  $x^{8}+2x^{4}-4x^{3}+x^{2}-1$
• Answer: -4

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