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**MTH 070 Elementary Algebra**

Chapter 5 – Exponents, Polynomials and Applications Section 5.3 – Introduction to Polynomials and Polynomial Functions Copyright © 2010 by Ron Wallace, all rights reserved.

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**Vocabulary Term Coefficient of a Term Degree of a Term A number: 17**

A variable: x A product: -2x3 Positive exponents only! No addition, subtraction, or division! Coefficient of a Term The “largest” constant factor of a term. That is, the number part of the term What about x, -x, x5, etc.? Degree of a Term The number of variable factors of the term.

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**Vocabulary Polynomial Leading Term Leading Coefficient**

A term or sum of terms. Note: Subtraction is considered adding the opposite. Convention: Put terms in order by degree. Leading Term Term of a polynomial with highest degree. Leading Coefficient Coefficient of the leading term. Degree of the Polynomial Degree of the leading term.

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**Vocabulary Monomial Binomial Trinomial A polynomial with one term. 5x3**

A polynomial with two terms. 3x2 – 5 Trinomial A polynomial with three terms. x2 – 4x + 3

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**Polynomial Expressions**

For the above polynomial, determine … The number of terms: The degree of the second term: The degree of the polynomial: The leading coefficient: The coefficient of the second term: The coefficient of the linear term: Is this polynomial a monomial, binomial or trinomial?

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**Evaluating a Polynomial**

Given a polynomial and a value for its variable … substitute the value for the variable and do the arithmetic. Example 1: Determine the value of x2 – 4x + 3 when x = 2 Example 2: Determine the value of x2 – 4x + 3 when x = –2

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**A Little Trick for Evaluating Polynomials**

Determine the value of x2 – 4x + 3 when x = –2

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**A Little Trick for Evaluating Polynomials**

Determine the value of x2 – 4x + 3 when x = –2

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**A Little Trick for Evaluating Polynomials**

Determine the value of x2 – 4x + 3 when x = –2

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**A Little Trick for Evaluating Polynomials**

Determine the value of x2 – 4x + 3 when x = –2

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**A Little Trick for Evaluating Polynomials**

Determine the value of x2 – 4x + 3 when x = –2

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**A Little Trick for Evaluating Polynomials**

Determine the value of x2 – 4x + 3 when x = –2

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**A Little Trick for Evaluating Polynomials**

Determine the value of 5x3 - 7x2 - 2 when x = 3

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**Functions … a review from 3.6**

Function … a named expression that gives only one result for each value of the variable. Notation: f(x) = an-expression-using-x Read as “f of x equals …” Doesn’t have to be f … g(x); h(x); p(x) … Doesn’t have to be x … f(a); g(m); d(t) … Evaluating a Function f(3) means replace the variable in the expression with 3 and do the arithmetic.

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Polynomial Functions A polynomial function is a function where the expression is a polynomial. Example: P(x) = 2x2 – 4x + 3 Linear Function Polynomial function of degree 1 Quadratic Function Polynomial function of degree 2 Cubic Function Polynomial function of degree 3

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Polynomial Functions If P(x) = 2x2 + 4x + 3, find P(0) & P(1) & P(–5)

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**Adding Polynomials “Combine Like Terms” Order of terms in the answer?**

i.e. Terms with the same variables can be combined by adding their coefficients. Order of terms in the answer? Descending order by degree Ascending order by degree Match the problem!

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Adding Polynomials Example …

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**Review: Subtracting Signed Numbers**

“Add the Opposite” a – b = a + (–b) Opposite? The number the same distance from zero on the other side of zero. –(5) = –5 –(–5) = 5 Essentially, multiplication by –1

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**Opposite of a Polynomial**

If p(x) is a polynomial, then its opposite is … –p(x) = (-1)p(x) Example: –(3x – 4) = (–1)(3x – 4) = –3x + 4 That is: Change the sign of every term.

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**Subtracting Polynomials**

If p(x) and q(x) are polynomials, then p(x) – q(x) = p(x) + (–q(x)) i.e. Add the opposite of the polynomial that follows the subtraction sign!

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**Subtracting Polynomials**

Example …

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Expressions with only multiplication, division and exponents are called monomials Write 3 monomials.

Expressions with only multiplication, division and exponents are called monomials Write 3 monomials.

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