# 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

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

• 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: $(x+i \sqrt{3})(x-i \sqrt{3})$