Find a polynomial function that has the given zeros. (There are many correct answers.) 4,2+√(7),

Find a polynomial function that has the given zeros. (There...

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Key Concepts

Constructing a Polynomial from its Zeros

To construct a polynomial from its given zeros, one uses the Factor Theorem by writing the polynomial as a constant multiplied by the product of factors corresponding to each zero. This method guarantees that the polynomial will have the desired zeros when expanded.

Conjugate Root Theorem

The Conjugate Root Theorem applies to polynomials with real coefficients and states that if an irrational or complex number is a zero of the polynomial, then its conjugate is also a zero. This ensures that the non-real parts cancel out, keeping the coefficients real.

Zeros of a Polynomial

Zeros of a polynomial are the values for which the polynomial evaluates to zero. These values are the solutions to the equation formed when the polynomial is set equal to zero. They indicate where the graph of the function crosses or touches the x-axis.

Factor Theorem

The Factor Theorem states that if a polynomial f(x) has a zero at x = r, then (x - r) is a factor of the polynomial. This theorem provides a method to construct a polynomial by multiplying its linear factors corresponding to each zero.

Transcript

00:02 We're going to find a polynomial function that has a given zeros of four, two plus a square to seven, and two minus a square to seven.

00:10 So that means x equals four, x equals two plus a square to seven, and x equals two minus the square to seven.

00:19 To help me find my factors, i'm going to make this so that i have equals zero.

00:25 So to do that, i'm going to subtract four from both sides on this one.

00:28 We get x minus four equals zero.

00:31 On this middle one, i'm going to say minus two minus a square to seven.

00:37 And i'm going to do that to both sides.

00:39 And we get x minus two minus the square to seven equals zero.

00:44 And over here, i'm going to subtract two and add the square to seven.

00:50 Subtract two and add the square to seven.

00:52 And we get x minus two plus the square to seven equals zero.

00:57 And these are my factors for this function.

00:59 So we get f of x equals x minus four.

01:03 Times x minus 2 minus the square to seven times x minus 2 plus the square to 7.

01:12 When we expand this out, we get x minus 4 times x squared minus 2x plus the square to 7x minus 2x plus 4 minus 2 square roots of 7 minus the square width to 7 minus the square to 7x plus 2 square to 7x plus 2 square width of 7.

01:45 Minus seven.

01:46 And there's a lot going on there...

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