Writing Polynomial Functions With Given Zeros | Precalculus

The Organic Chemistry Tutor

14 Feb 201817:00

EducationalLearning

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TLDRThis video tutorial provides a step-by-step guide on constructing polynomial functions from their zeros. It begins with a quadratic function example, explaining how to use given zeros (1, 2) and a point (x=3, f(x)=6) to derive the polynomial 3x^2 - 9x + 6. The process is then applied to higher degree polynomials, including a cubic function with real and imaginary zeros, leading to the function 2x^3 - 6x^2 + 8x - 24, and a quartic function with complex zeros, resulting in 15/2(4x^4 - 45x^3 + 1200x^2 - 1275x - 1305). The video demonstrates factoring, foiling, and calculating the leading coefficient to ensure the polynomial meets specified characteristics.

Takeaways

📚 The video teaches how to construct polynomial functions given their zeros and additional function values.
🔍 For a quadratic function with zeros at 1 and 2, and a value of 6 when x=3, the polynomial is derived step by step.
📐 The leading coefficient 'a' is determined by substituting x and the function's value into the factored form and solving for 'a'.
📈 The standard form of the polynomial is obtained by expanding the factored form and combining like terms.
🎓 The process is demonstrated for a quadratic polynomial, resulting in the function 3x^2 - 9x + 6.
🌟 The video extends the method to cubic and quartic polynomials, including those with complex zeros.
🧩 For polynomials with complex zeros, pairs of complex conjugates are used, and i^2 is treated as -1 during the expansion.
📉 The leading coefficient for higher-degree polynomials is found by substituting a given x-value and the corresponding function value.
📝 The video provides a detailed example of constructing a cubic polynomial with zeros at 3, 2i, -2i, and a value of 40 at x=4, resulting in 2x^3 - 6x^2 + 8x - 24.
🔢 Another example constructs a quartic polynomial with real and imaginary zeros, and a value of 900 at x=3, leading to a complex polynomial in standard form.
📚 The final quartic polynomial example results in a function of 30x^4 - 675x^3 + 1200x^2 - 1275x - 1305, showcasing the method's application to higher degrees.

Q & A

What is the degree of a polynomial function if its zeros are one and two, and it has a value of six when x is three?
-The degree of the polynomial function is two, as it is determined by the number of zeros it has, which in this case are one and two.
How do you find the factors of a polynomial with zeros one and two?
-The factors are found by setting x equal to each zero and moving them to the left side of the equation, resulting in (x - 1) and (x - 2).
What is the leading coefficient 'a' of the polynomial with zeros one and two, given that the function has a value of six when x is three?
-To find 'a', substitute x with three and the function with six, then divide the resulting value by the product of the factors evaluated at x=3, which is 2, resulting in a = 3.
What is the standard form of the polynomial function with zeros one and two and a value of six when x is three?
-The standard form is obtained by multiplying the factors and combining like terms, resulting in 3x^2 - 9x + 6.
What is the significance of having imaginary zeros come in pairs in polynomial functions?
-Imaginary zeros come in pairs because when you multiply complex conjugates, the imaginary parts cancel out, leaving a real coefficient, which is necessary for a polynomial with real coefficients.
How do you find the factors of a polynomial with zeros three, 2i, and -2i?
-The factors are found by setting x equal to each zero and moving them to the left side of the equation, resulting in (x - 3), (x - 2i), and (x + 2i).
What is the process to multiply complex conjugate zeros in a polynomial?
-To multiply complex conjugate zeros, you multiply the real parts and the imaginary parts separately, and since i^2 = -1, the imaginary parts will cancel out, leaving a real number.
How do you determine the leading coefficient 'a' of a cubic polynomial with zeros three, 2i, and -2i, given that the function has a value of 40 when x is four?
-To find 'a', substitute x with four and the function with 40, then divide the resulting value by the product of the factors evaluated at x=4, which is 20, resulting in a = 2.
What is the standard form of the cubic polynomial with zeros three, 2i, and -2i and a value of 40 when x is four?
-The standard form is obtained by multiplying the factors and combining like terms, resulting in 2x^3 - 6x^2 + 8x - 24.
How do you handle a polynomial with a degree of four and given zeros and a specific value at a certain x?
-You first write the factors based on the zeros, then multiply the factors and combine like terms. Use the given value at a specific x to find the leading coefficient 'a'.
What is the final polynomial function with a degree of four, zeros two, -3/4, 5-2i, and 5+2i, and a value of 900 when x is three?
-The polynomial function is 15/2 * (4x^4 - 5x^3 - 6x^2) * (x^2 - 10x + 29), which simplifies to 30x^4 - 675/2x^3 + 1200x^2 - 1275x - 1305.

Outlines

00:00

📚 Writing Polynomial Functions from Zeros

This paragraph introduces the concept of constructing polynomial functions given their zeros and an additional point. The example provided is a quadratic polynomial with zeros at x=1 and x=2 and a value of 6 when x=3. The process involves moving the zeros to the left side to form factors (x-1) and (x-2) and determining the leading coefficient 'a' by substituting x=3 into the equation. The final polynomial, after simplifying, is 3x^2 - 9x + 6. The paragraph also sets up the next example involving a cubic polynomial with real and imaginary zeros and a specific function value.

05:02

🔍 Constructing a Cubic Polynomial with Real and Imaginary Zeros

The second paragraph delves into constructing a cubic polynomial with zeros at x=3, 2i, and -2i, and a function value of 40 when x=4. It explains that imaginary numbers come in pairs and shows the conversion of zeros into factors. The polynomial is represented as a product of factors with an unknown leading coefficient 'a'. By substituting x=4 and simplifying, the leading coefficient is determined to be 2. The final polynomial is given in both factored and standard forms, resulting in 2x^3 - 6x^2 + 8x - 24.

10:02

🎓 Writing a Quartic Polynomial with Mixed Zeros

The third paragraph focuses on creating a quartic polynomial with zeros at x=2, -3/4, 5-2i, and 5+2i, and a function value of 900 when x=3. It outlines the steps to convert the zeros into factors and simplify the complex factors. The polynomial is expressed as a product of these factors with an unknown 'a'. By substituting x=3 and solving for 'a', the leading coefficient is found to be 15/2. The paragraph concludes with the polynomial in factored form, which is then expanded to its standard form.

15:04

📘 Detailed Expansion of a Quartic Polynomial

The final paragraph provides a detailed expansion of the quartic polynomial from the previous section. It breaks down the multiplication of trinomials and combines like terms to arrive at the standard form of the polynomial. The process involves careful arithmetic and algebraic manipulation to ensure accuracy. The resulting polynomial is given as 30x^4 - 675/2x^3 + 1200x^2 - 1275x - 1305, showcasing the complete solution to the problem.

Mindmap

Keywords

💡Polynomial functions

Polynomial functions are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In the video, polynomial functions are the central theme, as the script discusses how to construct them given certain characteristics like zeros and specific function values.

💡Zeros

In the context of polynomial functions, zeros refer to the values of the variable (usually x) that make the function equal to zero. The script explains how to use given zeros to start constructing a polynomial by creating factors like 'x - zero', which are then multiplied together to form the polynomial.

💡Degree of a polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It is a key characteristic that helps determine the general form of the polynomial. The video mentions polynomials of degrees two and three, indicating the highest power of x in the polynomial.

💡Factored form

The factored form of a polynomial is when it is expressed as a product of its factors, which are typically (x - zero) terms derived from the zeros of the polynomial. The video demonstrates converting the polynomial into its factored form as an intermediate step in the construction process.

💡Standard form

The standard form of a polynomial is when it is written as a sum of terms with descending powers of the variable, starting from the highest degree term down to the constant term. The script shows the transition from factored form to standard form by expanding and combining like terms.

💡Coefficient

A coefficient in a polynomial is the numerical factor that multiplies a term with a variable. In the video, determining the correct coefficient 'a' is crucial for constructing the polynomial correctly, as it scales the entire polynomial expression.

💡Foil method

The FOIL method is a technique used to multiply two binomials (expressions with two terms). It stands for First, Outer, Inner, Last, which refers to the terms you multiply together. The video uses the FOIL method to expand products of binomials when constructing the polynomial.

💡Imaginary numbers

Imaginary numbers are a category of numbers that, when squared, result in a negative number. They are represented by the letter 'i', where i^2 = -1. The script discusses polynomials with zeros that include imaginary numbers, emphasizing that they come in conjugate pairs.

💡Complex solutions

Complex solutions are solutions to equations that involve both real and imaginary numbers. In the context of the video, when zeros include imaginary numbers, the polynomial will have complex solutions, and the script shows how to handle these in the construction of the polynomial.

💡Combining like terms

Combining like terms is the process of adding or subtracting coefficients of terms that have the same variable raised to the same power. The video demonstrates this process as part of converting the polynomial from factored form to standard form.

💡Distributing

Distributing is the process of multiplying a term by each term inside a set of parentheses. In the video, distributing is used when multiplying a coefficient by a polynomial expression, as seen when determining the final form of the polynomial.

Highlights

Introduction to writing polynomial functions given zeros.

Example of a quadratic polynomial with zeros at one and two, and a value of six when x is three.

Method of converting zeros into factors for polynomial functions.

Determining the leading coefficient 'a' using a given function value.

Factored form of a quadratic polynomial derived from its zeros and a specific value.

Conversion of the factored form to the standard form of a polynomial.

Combining like terms to simplify the polynomial expression.

Second example of a cubic polynomial with complex zeros and a specific value.

Explanation of imaginary zeros coming in pairs in polynomial functions.

Foil method for multiplying binomials with complex numbers.

Simplification of polynomials using properties of 'i' squared.

Calculation of the leading coefficient 'a' for the cubic polynomial.

Expansion of the cubic polynomial using the foil method.

Third example of a quartic polynomial with mixed real and complex zeros.

Process of converting mixed zeros into polynomial factors.

Foil method for complex expressions resulting in a quadratic polynomial.

Calculation of the leading coefficient 'a' for the quartic polynomial.

Final polynomial function in standard form for the quartic example.

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Takeaways

Q & A

What is the degree of a polynomial function if its zeros are one and two, and it has a value of six when x is three?

How do you find the factors of a polynomial with zeros one and two?

What is the leading coefficient 'a' of the polynomial with zeros one and two, given that the function has a value of six when x is three?

What is the standard form of the polynomial function with zeros one and two and a value of six when x is three?

What is the significance of having imaginary zeros come in pairs in polynomial functions?

How do you find the factors of a polynomial with zeros three, 2i, and -2i?

What is the process to multiply complex conjugate zeros in a polynomial?

How do you determine the leading coefficient 'a' of a cubic polynomial with zeros three, 2i, and -2i, given that the function has a value of 40 when x is four?

What is the standard form of the cubic polynomial with zeros three, 2i, and -2i and a value of 40 when x is four?

How do you handle a polynomial with a degree of four and given zeros and a specific value at a certain x?

What is the final polynomial function with a degree of four, zeros two, -3/4, 5-2i, and 5+2i, and a value of 900 when x is three?