6.3 - Binomial Radical Expressions

6.3 - Binomial Radical Expressions

You remember multiplying two binomials, right?

•  $\left(x+2\right)\left(x-5\right)$  

•  $\left(x\right)\left(x\right)+\left(x\right)\left(-5\right)+\left(2\right)\left(x\right)+\left(2\right)\left(-5\right)$  

•  $x^2-5x+2x-10$  

•  $x^2-3x-10$  

You do the distributive property twice

What some of your teachers maybe called "FOIL" back in the day

We can take the same approach with radical binomials

Multiply and simplify  $\left(1+\sqrt{3}\right)\left(2+\sqrt{5}\right)$  

•  $\left(1\right)\left(2\right)+\left(1\right)\left(\sqrt{5}\right)+\left(\sqrt{3}\right)\left(2\right)+\left(\sqrt{3}\right)\left(\sqrt{5}\right)$  

•  $2+\sqrt{5}+2\sqrt{3}+\sqrt{15}$  

• Can this expression be simplified further?

• No. We can only add radicals when they have the same radical part

Multiply and simplify $\left(5-4\sqrt{5}\right)\left(-2+\sqrt{5}\right)$  

•  $\left(5\right)\left(-2\right)+\left(5\right)\left(\sqrt{5}\right)+\left(-4\sqrt{5}\right)\left(-2\right)+\left(-4\sqrt{5}\right)\left(\sqrt{5}\right)$  

•  $-10+5\sqrt{5}+8\sqrt{5}-20$  

•  $-30+13\sqrt{5}$  

You try

Multiple Choice

 Multiply and simplify $\left(5+4\sqrt{3}\right)\left(3+\sqrt{3}\right)$ 

•  $8+4\sqrt{3}$  

•  $20$  

•  $17\sqrt{3}$  

•  $27+17\sqrt{3}$  

Special Case: multiplying conjugates. Example: $\left(4+\sqrt{5}\right)\left(4-\sqrt{5}\right)$  

•  $\left(4\right)\left(4\right)\ +\left(4\right)\left(-\sqrt{5}\right)+\left(\sqrt{5}\right)\left(4\right)+\left(\sqrt{5}\right)\left(-\sqrt{5}\right)$  

•  $16-4\sqrt{5}+4\sqrt{5}-5$  

• The middle two terms are opposites, right? So they sum to zero.

• Then you're left with 16-5 = 11

You try

Multiple Choice

Multiply and simplify $\left(6+\sqrt{2}\right)\left(6-\sqrt{2}\right)$  

•  $6-2=4$  

•  $36+4=40$  

•  $36-2=34$  

• Cannot be simplified

What if there is a radical binomial in the denominator of a fraction?

• We have to rationalize the denominator

• We see that multiplying by a conjugate eliminates the radical term

• So we multiply the numerator and denominator by the conjugate

Simplify $\frac{4}{2-\sqrt{3}}$  

•  $\frac{4\cdot\left(2+\sqrt{3}\right)}{\left(2-\sqrt{3}\right)\cdot\left(2+\sqrt{3}\right)}$  

•  $\frac{4\left(2+\sqrt{3}\right)}{4-3}$  

•  $4\left(2+\sqrt{3}\right)\ or\ 8+4\sqrt{3}$  

You try

Multiple Choice

 $\frac{5}{3-\sqrt{6}}$  

Simplify

•  $-\frac{5}{3}$  

•  $\frac{5\left(3+\sqrt{6}\right)}{3}$  

•  $15-\sqrt{30}$  

•  $\frac{15+\sqrt{6}}{6}$  

Poll

How do you feel about multiplying binomial radical expressions?

• I feel great, I understand everything!

• I feel pretty good, I didn't get everything right on here but I understand my mistakes.

• I feel okay, I would definitely need more practice if we were gonna have a quiz soon.

• I am struggling and I need more help.