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MTH 070 Elementary Algebra Chapter 5 – Exponents, Polynomials and Applications Section 5.3 – Introduction to Polynomials and Polynomial Functions Copyright.

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Presentation on theme: "MTH 070 Elementary Algebra Chapter 5 – Exponents, Polynomials and Applications Section 5.3 – Introduction to Polynomials and Polynomial Functions Copyright."— Presentation transcript:

1 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.

2 Vocabulary  Term A number: 17 A variable: x A product: -2x 3  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, x 5, etc.?  Degree of a Term The number of variable factors of the term.

3 Vocabulary  Polynomial 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.

4 Vocabulary  Monomial A polynomial with one term. 5x 3  Binomial A polynomial with two terms. 3x 2 – 5  Trinomial A polynomial with three terms. x 2 – 4x + 3

5 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?

6 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 x 2 – 4x + 3 when x = 2  Example 2: Determine the value of x 2 – 4x + 3 when x = –2

7 A Little Trick for Evaluating Polynomials Determine the value of x 2 – 4x + 3 when x = –2

8 A Little Trick for Evaluating Polynomials Determine the value of x 2 – 4x + 3 when x = –2

9 A Little Trick for Evaluating Polynomials Determine the value of x 2 – 4x + 3 when x = –2

10 A Little Trick for Evaluating Polynomials Determine the value of x 2 – 4x + 3 when x = –2

11 A Little Trick for Evaluating Polynomials Determine the value of x 2 – 4x + 3 when x = –2

12 A Little Trick for Evaluating Polynomials Determine the value of x 2 – 4x + 3 when x = –2

13 A Little Trick for Evaluating Polynomials Determine the value of 5x 3 - 7x when x = 3

14 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.

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

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

17 Adding Polynomials  “Combine Like Terms” 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!

18 Adding Polynomials  Example …

19 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

20 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.

21 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!

22 Subtracting Polynomials  Example …


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