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Chapter 5 Polynomials and Polynomial Func7ons Multiplying Polynomials
Essen%al Understandings • Use proper7es of exponents to evaluate and simplify expressions involving powers • Mul7ply polynomial func7ons Properties of Exponents (Multiplying)
Property
Rule
Example
Product of Powers
Power of a Power
Power of a Product
1 Simplifying Algebraic Expressions b) ( 3xy 2 )
a) 3xy ( − x 2 y )
c) ( 2x 2 z ) ( 5xy 2 )
3
3
2
Mul7plying Polynomials Perform the indicated opera7ons and simplify. Write answer in standard form. f ( x) = ( x + 2)( x − 2)( x − 3)
Perform the indicated opera7ons and simplify. Write answer in standard form. f ( x) = ( x + 2)( x + 2)( x − 2)( x − 3)
2 5-­‐1 Polynomial Func%ons Essen%al Understandings •  A polynomial func7on has dis7nguishing “behaviors.” You can look at its algebraic form and know something about its graph. You can look at its graph and know something about its algebraic form. Standard Form of a Polynomial Func%on The standard form of a polynomial func7on arranges the terms by degree in descending numerical order. P( x ) = 4 x 3 + 3 x 2 + 5 x − 2
You can classify a polynomial by its degree or its number of terms. Degree Name 0 1 2 3 4 5 Example # of Terms Term Name 3 Write each polynomial in standard form. What is the classifica7on of each polynomial by degree? By number of terms? 1) 3x + 9 x 2 + 5
2) 4 x − 6 x 2 + x 4 + 10 x 2 − 12
The degree of a polynomial func7on affects the shape of its graph and determines the maximum number of turning points, or places where the graph changes direc7ons. It also effects the end behavior, or the direc7ons of the graph to the far leT and to the far right. You can determine the end behavior of a polynomial func7on of degree n from the leading term axn of the standard form. End Behavior of a Polynomial Func%on a Posi7ve n Even n Odd Up and Up Down and Up a Nega7ve Down and Down Up and Down 4 What is the end behavior of the graph? 1) y = 4 x3 − 3x
2) y = −2 x 4 + 8 x3 − 8 x 2 + 2
• The graph of a polynomial func7on of degree n has at most n-­‐1 turning points. • The graph of a polynomial func7on of odd degree has an even number of turning points. • The graph of a polynomial func7on of even degree has an odd number of turning points. What is the graph of each cubic func7on? Describe the graph. y=
x 1 3
x
2
y 5 What is the graph of each cubic func7on? Describe the graph. y = − x3 + 2 x 2 − x − 2
x y By analyzing the differences of consecu7ve y-­‐
values, it is possible to determine the least-­‐degree polynomial func7on that could generate the data. If the first differences are constant, the func7on is linear, if the second differences (but not the first) are constant, the func7on is quadra7c. If the third differences (but not the second) are constant, the func7on is cubic, and so on. What is the degree of the polynomial func7on that generates the data shown. x y -­‐3 -­‐1 -­‐2 -­‐7 -­‐1 -­‐3 0 5 1 11 2 9 3 -­‐7 6 What is the degree of the polynomial func7on that generates the data shown. x y -­‐3 23 -­‐2 -­‐16 -­‐1 -­‐15 0 -­‐10 1 -­‐13 2 -­‐12 3 29 5-­‐2 Polynomials, Linear Factors, and Zeros Essen%al Understandings •  Finding the zeros of a polynomial func7on will help you factor the polynomials, graph the func7on and solve the related polynomial equa7ons. Wri%ng a Polynomial in Factored Form What is the factored form of x 3
− 2 x 2 − 15 x ?
7 What is the factored form of x3 − x 2 − 12 x ?
What are the zeros of y = ( x + 2)( x − 1)( x − 3)?
Graph the func7on. Make a table of x values beyond and between the zeros. x y What are the zeros of y = x( x − 3)( x + 5)
Graph the func7on. x y 8 Factor Theorem The expression (x-­‐a) is a factor of a polynomial if and only if the value of a is a zero of the related polynomial func7on. What is the cubic polynomial func7on in standard form with zeros -­‐2, 2, and 3? What is the quar7c polynomial func7on in standard form with zeros -­‐2,-­‐2, 2, and 3? f ( x) = ( x + 2)( x + 2)( x − 2)( x − 3) Can also be wriaen as f ( x) = ( x + 2)2 ( x − 2)( x − 3).
The repeated factor (x+2) makes -­‐2 a mul7ple zero. Since the factor appears twice, you can say that -­‐2 is a zero of mul7plicity 2. In general, a is a zero of mul7plicity n means that x-­‐a appears n 7mes. 9 If the graph of a polynomial function has
several turning points, the function can
have a relative maximum and a relative
minimum. A relative maximum is the value of the
function at an up-to-down turning point. A relative minimum is the value of the
function at the down-to-up turning point. 5-­‐3 Solving Polynomial Equa%ons Essen%al Understandings •  Knowing the zeros of a polynomial func7on gives informa7on about its graph. Polynomial Factoring Techniques Techniques Examples Factoring out the GCF Factor out the greatest common factor of all the terms. Quadra7c Trinomials For ax2+bx+c, find factors with product of ac and sum of b. Perfect Square Trinomials 2
2
2
a + 2ab + b = (a + b)
2
2
2
a − 2ab + b = (a − b)
Difference of Squares 2
2
a − b = (a + b)(a − b)
10 Polynomial Factoring Techniques Cont. Techniques Examples Factoring by Grouping ax + ay + bx + by
= a( x + y ) + b( x + y )
= (a + b)( x + y )
Sum or Difference of Cubes a3 + b3 = (a + b)(a 2 − ab + b2 )
3 3
a − b = (a − b)(a 2 + ab + b 2 )
Solving Polynomial equa%ons by FACTORING What are the real or imaginary solu7ons of the polynomial equa7on 2 x3 − 5 x 2 = 3x ?
What are the real or imaginary solu7ons of the polynomial equa7on 3x 4 + 12 x 2 = 6 x3 ?
11 What are the real or imaginary solu7ons of the polynomial equa7on x 4 − 3x 2 = 4?
What are the real or imaginary solu7ons of the polynomial equa7on? ( x 2 − 1)( x 2 + 4) = 0
x5 + 4 x3 = 5 x 4 − 2 x3
5-­‐4 Dividing Polynomials Objec%ves: • Divide polynomials by using long division. 12 Dividing Polynomials Numerical long division and polynomial long division are similar. 21 672
Dividing Polynomials 2x + 1 6x2 + 7 x + 2
When you divide a polynomial f (x) by a divisor d(x), you get a quo7ent polynomial q(x) and a remainder polynomial r(x). This is wriaen as f ( x)
r ( x)
= q ( x) +
d ( x)
d ( x)
f ( x) = d ( x)q ( x) + r ( x)
The degree of the remainder must be less than the degree of the divisor. 13 Dividing Polynomials ( 4x
2
+ 23x −16) ÷ ( x + 5)
Dividing Polynomials ( 3x
2
− 29 x + 56) ÷ ( x − 7 )
Is x 2 +
1 a factor of 3 x 4 − 4 x3 + 12 x 2 + 5?
14 Synthe%c Division Synthe7c division simplifies the long division process for dividing by a linear expression x-­‐a. To use synthe7c division, write the coefficients (including zeros) of the polynomial in standard form and omit all variables and exponents. For the divisor reverse the sign. Use synthe7c division to divide by . What is the quo7ent and remainder? Step 1: Reverse the sign of +2. Write the coefficients of the polynomial. Step 2: Bring down the first coefficient. Step 3: Mul7ply the coefficient by the divisor. Add to the next coefficient. Step 4: Con7nue mul7plying and adding through the last coefficient. Use synthe7c division to divide by . What is the quo7ent and remainder? 15 The Remainder Theorem If you divide a polynomial P(x) of degree by x-a, then the remainder is P(a). The remainder theorem provides a quick way to find the remainder of a polynomial. Given that what is P(3)? Given that P(-4)? what is 5-­‐5 Theorems About Roots of Polynomial Equa%ons Objec%ves: • The factors of a polynomial func7on can be used to solve the polynomial. 16 The factors of the numbers a n and a 0 in P( x) = an x n + an −1 x n −1 + ... + a1 x + a 0 can help you factor the polynomial and solve the equa7on P( x. ) =
0 . One way to find a root of the polynomial is to guess and check but this is inefficient unless there is a way to minimize the number of guesses. Finding a Ra%onal Root What are the ra7onal roots of f ( x) = x − 4 x − 11x + 30
The only possible ra7onal roots have the form: factors of constant term
factors of leading coefficient
3
2
17 Finding a Ra%onal Root 3
2
What are the ra7onal roots of f ( x) = x − 4 x − 11x + 30
Step 1: Iden7fy your possible ra7onal roots. Possible ra7onal roots: Step 2: Test each possible ra7onal root using synthe7c division un7l you find a root Step 3: Once you get a zero as a remainder, write out the new factor of the polynomial. Step 4: Con7nue finding roots and dividing un7l you have a second-­‐ degree polynomial. Use factoring methods or the quadra7c formula to find the remaining root. 5)Find the zeros by leing f (x) = 0. (Don’t forget to include the zeros you find using the ra7onal root theorem) Find all the real zeros of f ( x) = x + x − 2 x − 2
3
2
If a complex number or an irra7onal number is a root of a polynomial equa7on with ra7onal coefficients, so is its conjugate. 18 Using the Conjugate Root Theorem A quar7c polynomial P(x) has ra7onal coefficients. If 2 and 1 + i are roots of P(x) = 0, what are the two other roots? What is the third degree polynomial func7on with ra7onal coefficients that has roots -­‐4 and 2i?
What is the quar7c polynomial func7on with ra7onal coefficients that has roots 2-­‐3i, 8, 2?
19 What does Descartes’ Rule of Signs tell you about the real roots of x 3 − x 2 + 1 = 0?
What does Descartes’ Rule of Signs tell you about the real roots of 2 x 4 − x 3 + 3 x 2 − 1 = 0?
5-­‐6 The Fundamental Theorem of Algebra Objec%ves: • Use the fundamental Theorem of Algebra to solve polynomial equa7ons with complex solu7ons. 20 What are all the roots of x 5
− x 4 − 3x 3 + 3x 2 − 4 x + 4 = 0?
What are the zeros of f ( x) = x + x − 7 x − 9 x − 18?
4
3
2
5-­‐7 The Binomial Theorem Objec%ves: • Expand a binomial using Pascal’s Triangle. 21 To expand the power of a binomial in general, first mul7ply as needed. Then write the polynomial in standard form. ( a + b)3
Consider the expansions of (a + b) n for the first few values of n. The coefficients only column matches the numbers in Pascal’s Triangle. It is a triangular array of numbers in which the first and last number of each row is 1. Each of the other numbers in the row is the sum of the two numbers above it. 6
What is the expansion of (a + b) ?
22 8
What is the expansion of (a + b) ?
When you use the Binomial Theorem to expand ( x − 2)
a = x and b = -­‐2. 4
5
What is the expansion of (3
x − 5) ?
5-­‐9 Transforming Polynomial Func%ons Objec%ves: • Apply transforma7ons to graphs of polynomials. 23 Transforming a Polynomial Func%on y = a ( x − h ) + k represents all the cubic func7ons you 3
can obtain by stretching, compressing, reflec7ng or 3
transla7ng the cubic parent func7on y = x
What is the equa7on of the graph of y = x
3 under a ver7cal compression by the factor ½ followed by a reflec7on across the x-­‐axis, a horizontal transla7on 3 units to the right, and then a ver7cal transla7on 2 units up? Finding Zeros of a Cubic Func%on 3
What are all the real zeros of the func7on y = 3( x − 1) + 6?
What are all the real zeros of the func7on y
=
1
3
( x − 2 ) − 3?
2
24 Construc%ng a Quar%c Func%on with Two Real Zeros What is a quar7c func7on with only two real zeros, x = 5 and x = 9? What is a quar7c func7on with only two real zeros, x = 0 and x = 6? y = x( x − 6) ⋅ Q ( x )
y = x( x − 6)( x 2 + 1)
y = ( x 2 − 6 x )( x 2 + 1)
y = x 4 − 6 x3 + x 2 − 6 x
If the exponent b in y = ax
b is a posi7ve integer, the func7on is also a monomial func7on. If y = ax
b describes y as a power func7on of x, then y varies directly with the bth power of x. 25 Wind farms are a sources of renewable energy found around the world. The power P (in kilowaas) generated by a wind turbine varies directly as the cube of the wind speed v (in meters per second). The picture shows the power output of one turbine at one wind speed. To the nearest kilowaa, how much power does this turbine generate in 10 m/s wind? 26