In Exercises 61–62, find an nth-degree polynomial function with real coefficients satisfying the

In Exercises 61–62, find an nth-degree polynomial function...

Key Concepts

Using Function Value to Determine Coefficients

When constructing a polynomial with given zeros, there is often an undetermined scaling factor or leading coefficient. Specific function values provided in the problem, such as the value at a certain input, are used to solve for this unknown constant. By substituting the coordinates into the polynomial expression, one can determine the constant that ensures the polynomial meets all of the given conditions.

Multiplicity of Zeros

The multiplicity of a zero refers to the number of times a particular zero is repeated as a solution to the polynomial equation. A zero of higher multiplicity generally causes the graph of the polynomial to touch the axis and turn around rather than crossing it, affecting the shape and behavior of the graph near that point.

Polynomial Function

A polynomial function is an expression consisting of variables and coefficients combined using operations of addition, subtraction, multiplication, and non-negative integer exponents. The degree of a polynomial is determined by the highest power of the variable present, and it influences the overall behavior of the function, including the number of possible zeros and the end behavior of its graph.

Real Coefficients

When a polynomial function has real coefficients, any non-real (complex) zeros must occur in conjugate pairs. This means that if the polynomial has a complex zero, its complex conjugate is also a zero. This property is a direct consequence of the operations on complex numbers and ensures that the polynomial's output remains real for real inputs.

Complex Conjugate Zero Theorem

The Complex Conjugate Zero Theorem states that for any polynomial with real coefficients, if a complex number is a zero of the polynomial, then its conjugate is also a zero. This theorem is fundamental when constructing polynomials with specified complex zeros as it guarantees that the product of the factors associated with the complex zeros will result in a quadratic factor with real coefficients.

Transcript

00:01 We are told that x equals i is a zero for this problem, and we are told that x equals 3i is a zero for this problem.

00:11 But we know this is a fourth degree function, so there must be two other zeros that we aren't seeing here.

00:17 That's okay though, because since both these zeros are imaginary, we know they both have pairs, meaning if x equals i is a zero, then we must also have a zero of x equals negative i.

00:28 If x equals 3i is a zero, then we must also have a zero of x equals negative 3i.

00:34 That's four different zeros.

00:36 So now we can set up our linear factorization theorem, saying that f of x is equal to a sub n times the factors x minus i and x plus i, and times the factors x minus 3i and x plus 3i.

00:56 And now we just have to multiply all those factors together.

00:59 Now we can't do them all at once, so what i'm going to do is i'm going to foil these two together, and i'm going to foil these two together.

01:09 And then we'll take their two answers and multiply them together after that.

01:12 Okay, so we've got f of x equals a sub in, x times x is x squared, x times i is positive i x, negative i times x is negative i x, negative i times positive i is negative i squared on our other two x times x is x squared x times 3 i is positive 3 i x negative 3 i times x is negative 3 i x negative 3 i times positive 3 i would be negative 9 i squared now we should probably clean both of those up because otherwise it's a lot more multiplying than we have to do for our next step.

02:11 I see a positive ix and a negative ix that cancel.

02:15 Same with positive 3 ix and negative 3 ix.

02:18 Remember also that i squared is equivalent to negative 1.

02:23 Meaning we've got f of x equals a sub n times the quantity x squared plus 1 because it would be minus a negative 1.

02:36 Times the quantity x squared plus 9 because it'd be negative 9 times negative 1 right and so now we can go about trying to foil these two together so we've got f of x is equal to a sub n x squared times x squared would be x to the 4th x squared x squared times 9 would be 9 x squared 1 times x squared would be x squared 1 times 9 would be 9.

03:17 So we can clean this up a little bit then.

03:20 F of x is going to be equal to a sub n times x to the 4th plus 10 x squared plus 9...

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