1.6 Solving Polynomial Equations

• The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has at least one complex root
• The Fundamental Theorem of Algebra guarantees the exact number of distinct roots of a polynomial.

  False
• If a polynomial equation with real coefficients has a complex root $z =$$a +$$bi$, then its complex conjugate z = a - bi</latex> is also a root
• Complex conjugate roots occur only in pairs.
• Match the property of complex conjugate roots with its description:

  Occur in pairs ↔️ Always come in $a +$$bi$ and $a - bi$ form

  Apply only to polynomials with real coefficients ↔️ Complex coefficients must cancel out

  Enable factoring into quadratic factors with real coefficients ↔️ Allows simplifying polynomials
• The quadratic formula is $x =$$\frac{ - b \pm \sqrt{b^{2} - 4ac}}{2a}$, where $b^{2} - 4ac$ is the discriminant
• Complex roots of a quadratic equation appear when the discriminant is negative.
• Steps to solve the quadratic equation $x^{2} +$$2x +$$5 =$$0$:


  1️⃣ Identify $a =$$1$, $b =$$2$, $c =$$5$

  2️⃣ Calculate the discriminant $b^{2} - 4ac =$$- 16$

  3️⃣ Find the complex roots $x =$$- 1 \pm 2i$
• If $z =$$a +$$bi$ is a root of a polynomial equation with real coefficients, then its complex conjugate $\overline{z} =$$a - bi$ is also a root
• Complex conjugate roots always occur in pairs for polynomials with real coefficients.
• What does the Fundamental Theorem of Algebra state about non-constant polynomials with complex coefficients?

  Every non-constant polynomial has at least one complex root
• If $z =$$a +$$bi$ is a root of a polynomial equation with real coefficients, then its complex conjugate $\overline{z} =$$a - bi$ is also a root.
• What condition must a polynomial satisfy for its roots to appear as complex conjugate pairs?

  Real coefficients
• Complex conjugate roots enable factoring into quadratic factors with real coefficients
• What are the roots of the equation x^{2} + 4 = 0</latex>?

  $\pm 2i$
• To solve quadratic equations of the form $ax^{2} +$$bx +$$c =$$0$, you can use the quadratic formula.
• If the discriminant $b^{2} - 4ac$ is negative, the roots of a quadratic equation are complex.
• What are the roots of the equation $x^{2} +$$2x +$$5 =$$0$?


  $- 1 \pm 2i$
• Steps to solve higher-degree polynomial equations with complex roots

  1️⃣ Identify a complex root

  2️⃣ Divide the polynomial by (z - \text{root})(z - \text{conjugate})</latex>

  3️⃣ Solve the resulting quadratic equation

  4️⃣ Combine all roots found
• What are the complex roots of the polynomial $P(z) =$$z^{3} - 5z^{2} +$$9z - 5$ given one root $z =$$2 +$$i$?


  $2 +$$i, 2 - i, 1$