Take Note Theorem: Conjugate Root Theorem

If P(x) is a polynomial with rational coefficients, then irrational roots of P ( x ) = 0 p open x close equals 0  that have the form a + b eh plus square root of b  occur in conjugate pairs. That is, if a + b eh plus square root of b  is an irrational root with a and b rational, then a − b eh minus square root of b  is also a root.

If P(x) is a polynomial with real coefficients, then the complex roots of P ( x ) = 0 p open x close equals 0  occur in conjugate pairs. That is, if a + b i eh plus b i  is a complex root with a and b real, then a − b i eh minus b i  is also a root.

Problem 3 Using the Conjugate Root Theorem to Identify Roots

A quartic polynomial P(x) has rational coefficients. If 2 square root of 2  and 1 + i 1 plus i  are roots of P ( x ) = 0 , p open x close equals 0 comma  what are the two other roots?

Since P(x) has rational coefficients and 0 + 2 0 plus square root of 2  is a root of P ( x ) = 0 , p open x close equals 0 comma  it follows from the Conjugate Root Theorem that 0 − 2 0 minus square root of 2  is also a root.

Since P(x) has real coefficients and 1 + i 1 plus i  is a root of P ( x ) = 0 , p open x close equals 0 comma  it follows that 1 − i 1 minus i  is also a root.

The two other roots are − 2 negative square root of 2  and 1 − i . 1 minus i .

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3. A cubic polynomial P(x) has real coefficients. If 3 − 2 i 3 minus 2 i  and 5 2 5 halves  are two roots of P ( x ) = 0 , p open x close equals 0 comma  what is one additional root?

Problem 4 Using Conjugates to Construct a Polynomial

Multiple Choice What is a third-degree polynomial function y = P ( x ) y equals p open x close  with rational coefficients so that P ( x ) = 0 p open x close equals 0  has roots − 4 negative 4  and 2i?

1. P ( x ) = x 3 − 2 x 2 − 16 x + 32 p open x close equals , x cubed , minus , 2 x squared , minus 16 x plus 32
2. P ( x ) = x 3 − 4 x 2 + 4 x − 16 p open x close equals , x cubed , minus , 4 x squared , plus 4 x minus 16
3. P ( x ) = x 3 + 4 x 2 + 4 x + 16 p open x close equals , x cubed , plus 4 , x squared , plus 4 x plus 16
4. P ( x ) = x 3 + 4 x 2 + 4 x − 16 p open x close equals , x cubed , plus 4 , x squared , plus 4 x minus 16

Since 2 i 2 i  is a root, then − 2 i negative 2 i  is also a root.

P ( x ) = ( x + 2 i ) ( x − 2 i ) ( x + 4 ) Write the polynomial function .   = ( x 2 + 4 ) ( x + 4 ) Multiply the complex conjugates .   = x 3 + 4 x 2 + 4 x + 16 Write the polynomial function in standard form . table with 3 rows and 3 columns , row1 column 1 , p , open x close , column 2 equals . open , x plus 2 i , close . open , x minus 2 i , close . open , x plus 4 , close , column 3 cap writethepolynomialfunction . . , row2 column 1 , , column 2 equals . open . x squared , plus 4 . close . open , x plus 4 , close , column 3 cap multiplythecomplexconjugates . . , row3 column 1 , , column 2 equals , x cubed , plus 4 , x squared , plus 4 x plus 16 , column 3 cap writethepolynomialfunctioninstandardform . . , end table

The equation x 3 + 4 x 2 + 4 x + 16 = 0 x cubed , plus 4 , x squared , plus 4 x plus 16 equals 0  has rational coefficients and has roots − 4 negative 4  and 2i. The correct answer is C.

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4. What quartic polynomial equation has roots 2 − 3 i , 8 , 2 ? 2 minus 3 i comma 8 comma 2 question mark