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5.7 – Polynomial Identities

Key Terms

  • Polynomial Identity – A polynomial expression set equal to a different representation of an equivalent polynomial expression.
    • Ex. (x-y)(x+y)=x^2-y^2
    • Ex. (ax-y)^2=a^2x^2-2axy+y^2

 


Review

  • Quadratic Formula
    • x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}

 

  • Rationalizing Denominators
    • ex. \frac{5}{\sqrt{7}+5}
    • Multiply both sides by the conjugate: \frac{5}{\sqrt{7}+5} \cdot \frac{\sqrt{7}-5}{\sqrt{7}-5}
    • Simplify: \frac{5 \sqrt{7}-25}{7-25}
    • Simplify: \frac{5 \sqrt{7}-25}{-18}

 


Notes

  • Polynomial Identities
    • are shortcuts
    • cut down on problem-solving time
    • show alternative forms of higher-order polynomials
    • some polynomial identities are trivial (very simple) and don’t help you at all

 

  • Difference of Cubes Identity
    • a^3-b^3=(a+b)(a^2+ab+b^2)

 

  • Sum of Cubes Identity
    • a^3+b^3=(a+b)(a^2-ab+b^2)

 

  • Two Squares Identity
    • (a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2

 

  • Difference of Squares Identity
    • is useful in finding the complex factors of the sum of two squares
    • a^2-b^2=(a+b)(a-b)
    • ex. 4x^2-25y^2=(2x+5y)(2x-5y)
    • ex. x^2+7
      • Remember: i= \sqrt{-1}
      • Difference of Squares: x^2-(i \sqrt{7})^2=(x+i \sqrt{7})(x-i \sqrt {7})

 

  • Pythagorean Triple Generator
    • (x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2

 


Examples

  • Ex 1. Which identity could be used to rewrite the expression x^9-8?
    • It can be written: (x^3-2)^3
    • So, Difference of Cubes identity

 

  • Ex 2. How would the expression (x^2+4)(y^2+4) be written using Two Squares?
    • Answer: (xy-4)^2+(2x+2y)^2

 

  • Ex 3. Which combination of integers can be used to generate the Pythagorean triple: (7, 24, 25)?
    • Answer: x = 4, y = 3
    • Use the Pythagorean Triple Generator: (x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2
    • Substitute: (4^2+3^2)^2=(4^2-3^2)^2+[(2)(4)(3)]^2
    • Simplify: (16+9)^2=(16-9)^2+(24)^2
    • Simplify some more: 25^2=7^2+24^2
    • So, the triple is 7, 24, 25

 

  • Ex 4. The Quadratic Formula gives which roots for the equation: 3x^2+3x=2?
    • Subtract 2 on both sides: 3x^2+3x-2=x
    • List a, b, and c: a = 3, b = 3, and c = -2
    • Substitute values into the quadratic formula: x=\frac{-3 \pm \sqrt{3^2-4(3)(-2)}}{2(3)}
    • Simplify: x=\frac{-3 \pm \sqrt{9+24}}{6}
    • Simplify some more: x=\frac{-3 \pm \sqrt{33}}{6}

 

  • Ex 5. Difference of Squares gives which complex factors for the expression: x^2+3?
    • Answer: (x+i \sqrt{3})(x-i \sqrt{3})

 


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