 # 6.1 Using Properties of Exponents What you should learn: Goal1 Goal2 Use properties of exponents to evaluate and simplify expressions involving powers.

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6.1 Using Properties of Exponents What you should learn: Goal1 Goal2 Use properties of exponents to evaluate and simplify expressions involving powers. Use exponents and scientific notation to solve real-life problems. 6.1 Using Properties of Exponents

Product of Powers Property ex)

The Power of a Power Property ex)

Power of a Product ex)

Write each expression with positive exponents only. ex) Negative Exponents in Numerators and Denominators and ex)

Use the Zero-Exponent Rule ex) The Zero-Exponent Property ex)

Divide by using the Quotient Rule ex) The Quotient of Powers Property

Simplify by using the Quotient of Powers Rule ex) The Power of Quotient Property

ex) Simplify. ex)

Simplify. ex)

Reflection on the Section Give an example of a quadratic equation in vertex form. What is the vertex of the graph of this equation? assignment

6.2 Evaluating and Graphing Polynomial Functions What you should learn: Goal1 Goal2 Evaluate a polynomial function Graph a polynomial function. 6.2 Evaluating and Graphing Polynomial Functions

Polynomial- is a single term or sum of two or more terms containing variables in the numerator with whole number exponents. or

Polynomial- is a single term or sum of two or more terms containing variables in the numerator with whole number exponents. It is customary to write the terms in the order of descending powers of the variables. This is Standard Form of a polynomial.

Monomials-polynomials with one term. Example) 6 or 2x or Binomials-polynomials with two terms Example) Trinomials-polynomials with three terms. Example)

The Degree of If a does not equal zero, then the degree of is n. The degree of a nonzero constant is 0. no The constant “ 0 “ has no defined degree.

Polynomial Degree of the polynomial is the largest degree of its terms. Example) 2x, has a degree of 1 Example), has a degree of 2, has a degree of 3 Degree of the number is the exponent of the variable.. Example)

Classifying polynomials by degree Constant, Linear, Quadratic, Degree 0, Degree 1, Degree 2, Degree 3, Degree 4, Monomial Binomial Trinomial Monomial Polynomial Cubic, Quartic,

Directions: Use Direct Substitution to evaluate the Polynomial Function for the given value of x. f (x) =, when x = 3 f (3) = Make the Substitution. Goal1 Evaluate a polynomial function

Synthetic Substitution 1. Arrange polynomials in descending powers, with a 0 coefficient for any missing term. Directions: Use Synthetic Substitution to evaluate the Polynomial Function for the given value of x. Another way to evaluate a polynomial function is to use Synthetic Substitution. NOTICE

Synthetic Substitution

3 x-value Polynomial in standard form 2 0-85-7 2 61035 98 61830105 multiply add

Reflection on the Section Which term of a polynomial function is most important in determining the end behavior of the function? assignment

6.3 Adding, Subtracting, and Multiplying Polynomials What you should learn: Goal1 Add, subtract, and multiply polynomials 6.3 Adding, Subtracting, and Multiplying

Add or subtract as indicated ex)

Add or subtract as indicated ex)

Add or subtract as indicated (vertically) ex) (+)

Add or subtract as indicated (vertically) ex) (-)

ex) Use a vertical format to find each product +

Multiplying Monomials To multiply monomials, multiply the coefficients and then multiply the variables. Use the product rule for exponents to multiply the variables: Keep the variable and add the exponents. ex) multiply the coefficients and multiply the variables

ex)

Finding the product of the monomial and the polynomial ex)

Finding the product when neither is a monomial ex)

Multiply by using the rule for finding the product of the sum and difference ex) The Product of the Sum and Difference of Two Terms

Multiply by using the rule for the Square of a Binomial. ex) The Product of the Sum of Two Terms

Multiply by using the rule for the Square of a Binomial. ex) The Product of the Difference of Two Terms

Using the FOIL Method to Multiply Binomials ex)

Find the Product ex)

Find the Product ex)

Find the Product ex)

Find the Product ex)

Reflection on the Section How do you add or subtract two polynomials? assignment

6.4 Factoring and Solving Polynomial Equations What you should learn: Goal1 Goal2 Factor polynomial expressions 6.4 Factoring and Solving Polynomial Equations Use Factoring to solve polynomial expressions

Factoring Monomials means finding two monomials whose product gives the original monomial. Factoring is the process of writing a polynomial as the product of two or more polynomials. ex) Can be factored in a few different ways… a.) b.) c.) d.)

Find three factorizations for each monomial. Directions: 1.) 2.) 3.)

Find the greatest common factor. 1.) 2.) and GCF of 6 and 10 (or what # divides into 6 and 10 evenly) GCF of 6 and 10 (or what # divides into 6 and 10 evenly) When dealing with the variables, you take the variable with the smallest exponent as your GCF. and

Factoring out the greatest common factor. But, before we do that…do you remember the Distributive Property? When factoring out the GCF, what we are going to do is UN-Distribute.

Factor each polynomial using the GCF. ex)

Factor each polynomial using the Greatest Common Binomial Factor. ex)

Factor by Grouping Ex 1) Factor-out GCF from each binomial Factor-out GCF Factored by Grouping Group into binomials

Factoring the Sum or Difference of 2 Cubes 1.)Factoring the Sum of Two Cubes: 2.) Factoring the Difference of 2 Cubes:

Example 1) or Sum

Example2) or

Example 3) or Difference

Definition of a Quadratic Equation A quadratic equation in x is an equation that can be written in the standard form where a, b, and c are real numbers, with a = 0. A quadratic equation in x is also called a second-degree polynomial equation in x. /

The Zero-Product Principle If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero. If AB = 0, then A = 0 or B = 0.

example) According to the principle, this product can be equal to zero if either or +5 x = 5 +2 x = 2 The resulting two statements indicate that the solutions are 5 and 2.

example) Factor the Trinomial using the methods we know. or +1 x = 1/2 - 4 x = - 4 The resulting two statements indicate that the solutions are 1/2 and - 4. Solve a Quadratic Equation by Factoring (2x )(x ) = 0 - + 14 2x = 1

example) Move all terms to one side with zero on the other. Then factor. +3 The resulting two statements indicate that the solutions are 3. Solve a Quadratic Equation by Factoring (x )(x ) = 0 - - 33 x = 3 The trinomial is a perfect square, so we only need to solve once.

Reflection on the Section How can you use the zero product property to solve polynomial equations of degree 3 or more? assignment

6.5 The Remainder and Factor Theorems What you should learn: Goal1 Divide polynomials and relate the result to the remainder theorem and the factor theorem. 6.4 The Remainder and Factor Theorem

Divide using the long division ex) x + 7

Divide using the long division with Missing Terms ex)

Synthetic Division To divide a polynomial by x - c 1. Arrange polynomials in descending powers, with a 0 coefficient for any missing term. 2. Write c for the divisor, x – c. To the right, write the coefficients of the dividend. 31 4 -5 5

3. Write the leading coefficient of the dividend on the bottom row. 4. Multiply c (in this case, 3) times the value just written on the bottom row. Write the product in the next column in the 2 nd row. 31 4 -5 5 3 1 1 3

5. Add the values in the new column, writing the sum in the bottom row. 6. Repeat this series of multiplications and additions until all columns are filled in. 31 4 -5 5 3 1 1 3 3 7 add 7 21 add 16

7. Use the numbers in the last row to write the quotient and remainder in fractional form. The degree of the first term of the quotient is one less than the degree of the first term of the dividend. The final value in this row is the remainder. 1 4 -5 5 3 1 3 7 add 21 16 48 53

Synthetic Division To divide a polynomial by x - c 1 4 -2 Example 1) 1 3 -3 -5

Synthetic Division To divide a polynomial by x - c 21 0 -5 7 Example 2) 1 2 2 4 -2 5

Factoring a Polynomial Example 1) given that f(-3) = 0. -3 2 11189 -6-15-9 2 530 multiply Because f(-3) = 0, you know that (x -(-3)) or (x + 3) is a factor of f(x).

Factoring a Polynomial Example 2) given that f(2) = 0. 2 1 -2-918 2 0 -18 1 0-90 multiply Because f(2) = 0, you know that (x -(2)) or (x - 2) is a factor of f(x).

Reflection on the Section If f(x) is a polynomial that has x – a as a factor, what do you know about the value of f(a)? assignment

6.6 Finding Rational Zeros What you should learn: Goal1 Find the rational zeros of a polynomial. 6.6 Finding Rational Zeros

Find the rational zeros of 6.6 Finding Rational Zeros The Rational Zero Theorem solution List the possible rational zeros. The leading coefficient is 1 and the constant term is -12. So, the possible rational zeros are:

Find the Rational Zeros of 6.6 Finding Rational Zeros solution List the possible rational zeros. The leading coefficient is 2 and the constant term is 30. So, the possible rational zeros are: Example 1) Notice that we don’t write the same numbers twice

-21 7 -4 -28 Example 1) 1 -2 5 -10 -14 28 0 Use Synthetic Division to decide which of the following are zeros of the function 1, -1, 2, -2 x = -2, 2

11 4 1 -6 Example 1) 1 1 5 5 6 6 0 Find all the REAL Zeros of the function. x = -2, -3, 1

21 1 1 -9 -10 Example 2) 1 2 3 6 7 14 5 Find all the Real Zeros of the function. 10 0 1 3 7 5 1 2 -2 5 -5 0

x = 2, -1 1 3 7 5 1 2 -2 5 -5 0

Reflection on the Section How can you use the graph of a polynomial function to help determine its real roots? assignment

6.7 Using the Fundamental Theorem of Algebra What you should learn: Goal1 Use the fundamental theorem of algebra to determine the number of zeros of a polynomial function. 6.7 Using the Fundamental Theorem of Algebra THE FUNDEMENTAL THEOREM OF ALGEBRA If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers.

-51 5 -9 -45 Example 1) 1 -5 0 0 -9 45 0 Find all the ZEROs of the polynomial function. x = -5, -3, 3

Example 2) Find all the ZEROs of the polynomial function. 31 0 1 0 -12 1 3 3 9 10 30 90 0 NOT DONE YET

Example 1) Decide whether the given x-value is a zero of the function., x = -5 So, Yes the given x-value is a zero of the function.

Example 1) Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1. -4, 1, 5

Reflection on the Section How can you tell from the factored form of a polynomial function whether the function has a repeated zero? assignment At least one of the factors will occur more than once.

6.8 Analyzing Graphs of Polynomial Functions What you should learn: Goal1 Analyze the graph of a polynomial function. 6.8 Analyzing Graphs of Polynomial Functions Plot x-intercepts: Find the Turning Points: The y-coordinate of a turning point is a Local Maximum if the point is higher than all nearby points. The y-coordinate of a turning points is a Local Minimum if the point is lower that all nearby points.

Reflection on the Section Give an example of a quadratic equation in vertex form. What is the vertex of the graph of this equation? assignment

6.1 Using Properties of Exponents What you should learn: Goal1 Goal2 ghghhhghjghjghghggghjg hghjghjghjghjghjgjhb 6.1 Using Properties of Exponents

Reflection on the Section Give an example of a quadratic equation in vertex form. What is the vertex of the graph of this equation? assignment

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