Unit 3: Polynomial Functions By: Anika Ahmed, Pavitra Madala, and Varnika Kasu
Polynomial Function A polynomial function of degree n in standard form is where the a s are real numbers and the n s are nonnegative integers and an 0.
Polynomial Function A polynomial function of degree n in factored form is f(x)=a(x-)(x-)...(x-r) where the a s are real numbers.
Classifying Polynomials Monomial, Binomial, Trinomial, Polynomial
Classifying Polynomials A monomial has just one term. For example, 4x². Remember that a term contains both the variable(s) and its coefficient (the number in front of it.) So the is just one term. A binomial has two terms. For example: 5x² -4x. A trinomial has three terms. For example: 3y²+5y-2. Any polynomial with four or more terms is just called a polynomial. For example: 2y⁵+ 7y³- 5y²+9y-2.
Practice!! Practice classifying these polynomials by the number of terms: 1. 5y 2. 3x²-3x+1 3. 5y-10 4. 8xy 5. 3x⁴+x²-5x+9
Practice!! Practice classifying these polynomials by the number of terms: 1. 5y Monomial 2. 3x²-3x+1 3. 5y-10 4. 8xy 5. 3x⁴+x²-5x+9
Practice!! Practice classifying these polynomials by the number of terms: 1. 5y Monomial 2. 3x²-3x+1 Trinomial 3. 5y-10 4. 8xy 5. 3x⁴+x²-5x+9
Practice!! Practice classifying these polynomials by the number of terms: 1. 5y Monomial 2. 3x²-3x+1 Trinomial 3. 5y-10 Binomial 4. 8xy 5. 3x⁴+x²-5x+9
Practice!! Practice classifying these polynomials by the number of terms: 1. 5y Monomial 2. 3x²-3x+1 Trinomial 3. 5y-10 Binomial 4. 8xy Monomial 5. 3x⁴+x²-5x+9
Practice!! Practice classifying these polynomials by the number of terms: 1. 5y Monomial 2. 3x²-3x+1 Trinomial 3. 5y-10 Binomial 4. 8xy Monomial 5. 3x⁴+x²-5x+9 Polynomial
Application The profit P in millions of dollars for a t-shirt manufacturer can be modeled by P=-x³+4x²+x where x is the number of t-shirts produced in millions. currently, the company produces 4 million t-shirts and makes a profit of $4,000,000. What lesser number of tshirts could the company produce and still make the same profit?
Application Substitute P with 4 in P=-x³+4x²+x -x³+4x²+x = 4 -x³+4x²+x - 4 = 0 Since -x³+4x²+x - 4 = -x²(x - 4) + (x - 4) = (x - 4)(1 - x²) 1 - x² = 0 x² = 1 x = 1
Degree Polynomials
What is a Degree? A degree is THE HIGHEST EXPONENT!!
What is a Degree? A degree is THE HIGHEST EXPONENT!! Examples 1. 4x³-3x²+10
What is a Degree? A degree is THE HIGHEST EXPONENT!! Examples 1. 4x³-3x²+10 Degree=3
What is a Degree? A degree is THE HIGHEST EXPONENT!! Examples 1. 4x³-3x²+10 Degree=3 2. 10x
What is a Degree? A degree is THE HIGHEST EXPONENT!! Examples 1. 4x³-3x²+10 2. 10x Degree=3 Degree=1
What is a Degree? A degree is THE HIGHEST EXPONENT!! Examples 1. 4x³-3x²+10 2. 10x 3. 4 Degree=3 Degree=1
What is a Degree? A degree is THE HIGHEST EXPONENT!! Examples 1. 4x³-3x²+10 2. 10x 3. 4 Degree=3 Degree=1 Degree=0
What is a Degree? A degree is THE HIGHEST EXPONENT!! Examples 1. 4x³-3x²+10 2. 10x 3. 4 Degree=3 Degree=1 Degree=0 4. 7x⁵+9x⁶-x³+5x⁴-3x²+2+8x
What is a Degree? A degree is THE HIGHEST EXPONENT!! Examples 1. 4x³-3x²+10 2. 10x 3. 4 Degree=3 Degree=1 Degree=0 4. 7x⁵+9x⁶-x³+5x⁴-3x²+2+8x Degree=6
Linear Function A polynomial with a degree of 1 is called a linear function. It s form is. In this function, a and b are real numbers and a 0.
Quadratic Function A polynomial with a degree of 2 is called a quadratic function. It s form is. In this function, a, b, and c are real numbers and a 0.
Cubic Function A polynomial with a degree of 3 is called a cubic function. It s form is. In this function, a, b, c, and d are real numbers and a 0.
Quartic Function A polynomial with a degree of 4 is called a quartic function. It s form is. In this function, a, b, c, d, and e are real numbers and a 0.
Degree of 5 Function A polynomial with a degree of 5 is called a 5th degree function. It s form is. In this function, a, b, c, d, e, and f are real numbers and a 0.
AND SO ON. The next function is degree of 6, then is degree of 7, etc.
Even or Odd? Functions
Even or Odd? To determine whether a function is even or odd, you take the function and plug x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f ( x) = f (x), so all of the signs are the same), then the function is even. If you end up with the exact opposite of what you started with (that is, if f ( x) = f (x), so all of the signs are switched), then the function is odd. In all other cases, the function is a neither function.
Practice!! Determine algebraically whether f (x) = 3x² + 4 is even, odd, or neither The mirroring of the y-axis is an important clue which shows that a function is even. Also, the exponent on all of the terms are even. But to do this algebraically, I have to do: f ( x) = 3( x)² + 4 = 3(x²) + 4 = 3x² + 4 Since the result is the same as the original function, this function is even.
Practice!! Determine algebraically whether f (x) = 2x³ 4x is even, odd, or neither. The symmetry about the origin is very important when it comes to odd functions. Also, all of the exponents in this function are odd. To solve this algebraically f ( x) = 2( x)³ 4( x) = 2( x³) + 4x = 2x³ + 4x Since the result had the exact opposite of the original function, this function is odd.
Practice!! Determine algebraically whether f (x) = 2x³ 3x² 4x + 4 is even, odd, or neither. This function is not symmetrical about the y-axis or the origin. Also, the exponents in the function contain both even and odd integers. To solve this algebraically f ( x) = 2( x)³ 3( x)² 4( x) + 4 = 2( x³) 3(x²) + 4x + 4 = 2x³ 3x² + 4x + 4 The result didn t change all of the symbols, so it is not an odd function. The result isn t the exact same as the original function, so it is not an even function. Therefore, it is a neither function.
Multiplicities Functions
Multiplicity Def: The number of times a specific number is a solution for an equation. To find out the multiplicity of a function, simply look at the exponents outside of the parenthesis. For example, for the function f(x) = (x + 2)²(x 1)²: -2 has a multiplicity of 2 1 has a multiplicity of 2
Multiplicities in Graphs Multiplicity of 1 If the function crosses the x-axis at a point and doesn t have any turning points.
Multiplicities in Graphs Multiplicity of 2 If the function bounces off the x-axis at a certain point
Multiplicities in Graphs Multiplicity of 3 If the function takes some time on the x-axis and continues
End Behavior of Functions The end behavior of a graph describes the far left and the far right portions of the graph. Rise-Rise Fall-Rise Fall-Fall
Identifying Relative & Absolute Min. & Max. of a Graph
Division Long Division Synthetic Division
Long Division with Polynomials
Application The total number of video cassettes sold from 1995 to 2005 at Bob s store can be modeled by the function F(x)=4x³+14x²+200x+1560 and the number of kinds of video cassettes in Bob s store from 1995 to 2005 can be modeled by G(x) = 2x + 12, where x is the number of years since 1995. Using long division, find the average number of each kind of video cassettes that Bob sold.
Application
Synthetic Division k a b c d ka a r 1. Copy the leading coefficient. 2. Multiply horizontally. 3. Add vertically.
Synthetic Division
Application A box is to be mailed. The volume in cubic inches of the box can be expressed as the product of its three dimensions V(x) = x³-16x² +79x -120. The length is (x - 8). Find linear expressions for the other dimensions. Assume that the width is greater than the height.
Application 79 8 1-16 8 79-64 -120 120 (x-8)(x²-8x+15) (x-8)(x-5)(x-3) Width: (x-30 Height:(x-5) 1-8 15 0
Remainder Theorem
Factor Theorem
Transformation of f(x) Transformed Function Vertical shift of k units up y = f(x) + k, for k > 0 Vertical shift of k units down y = f(x) + k, for k < 0 Horizontal shift of h units right y = f(x - h), for h > 0 Horizontal shift of h units left y = f(x - h), for h < 0 Vertical stretch by a factor of a y = af(x), where a > 1 Vertical compression by a factor of a y = af(x), where 0 < a Horizontal stretch by a factor of 1/b y = f(bx), where 0 < b <1 Horizontal compression by a factor of 1/b y=f(bx), where b > 1 Reflection across the x--axis Reflection across the y--axis y = - f(x) y = f(-x)
Hope you learned something! AND GOOD LUCK ON THE EOG!