Concept 5 Rational expressions

Rational Expressions Rational Expression - the ratio of _____ polynomials. (Like a _____________ with a polynomial in the ________________ and the ______________________.  Examples: Non-Examples:  Characteristics: 1.   2. 2 fraction numerator denominator

Factor top & bottom: Cancel & Reduce if possible:

Factor top & bottom: Cancel & Reduce if possible:

Examples: 1. 45𝑥 2 15𝑥   2. 4𝑥 2 4𝑥+4 3. 𝑥 2 −4 𝑥+2 4. 14𝑥 2 +21𝑥 42𝑥 4

5. 3𝑥 3 + 7𝑥 2 6 𝑥 5 + 9𝑥 3   6. 𝑥 2 +9𝑥+18 4𝑥+24 7. 𝑥 2 +6𝑥−16 𝑥 2 +10𝑥−24 8. 𝑥 3 −𝑥 𝑥 3 − 7𝑥 2 +6𝑥 9. 4𝑥 6 + 24𝑥 3 12𝑥 2 −30 𝑥 4

Solving Polynomials Concept 6

To solve a polynomial, factor completely To solve a polynomial, factor completely. Any terms can then be set equal to zero and solve for x. 2. 6𝑥 2 +6𝑥=0 1. 𝑥 2 −𝑥 −42=0 6𝑥 𝑥+1 =0 𝑥−7 𝑥+6 =0 6𝑥=0 𝑥+1=0 𝑥−7=0 𝑥+6=0 𝑥=0 𝑥=−1 𝑥=7 𝑥=−6 3. 5𝑥 2 +37𝑥= −14 5𝑥 2 +37𝑥+14=0 5𝑥+2=0 5𝑥=−2 𝑥 2 +37𝑥+70=0 𝑥= −2 5 𝑥+2 𝑥+35 =0 5 5 5𝑥+2 𝑥+7 =0 𝑥+7=0 𝑥=−7

4. 7 𝑥 2 +53𝑥+28=0 5. 2𝑥 2 −4𝑥−3=−5 𝑥 2 +53𝑥+196=0 2𝑥 2 −4𝑥+2=0 4. 7 𝑥 2 +53𝑥+28=0 𝑥 2 +53𝑥+196=0 5. 2𝑥 2 −4𝑥−3=−5 𝑥+49 𝑥+4 =0 2𝑥 2 −4𝑥+2=0 7 7 2 𝑥 2 − 2𝑥+1 =0 𝑥+7 7𝑥+4 =0 2 𝑥−1 𝑥−1 =0 𝑥−1 =0 𝑥+7=0 7𝑥+4=0 𝑥=1 7𝑥=−4 𝑥=−7 𝑥= −4 7

Quadratic Formula – used to solve equations when it can’t be factored. Before substituting in values for a, b, and c, rearrange the equation to look like. 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 The quadratic formula is used to solve for values that make the equation true and are not factorable.

𝑥= − ± 2 −4 2 𝑥= − −8 ± −8 2 −4 1 −48 2 1 So a = = 1 b = = -8 𝑥= 8 ± 64+192 2 c = = -48 𝑥= 8 ± 256 2 1 -48 -8 -8 𝑥= 8 ±16 2 𝑥= 8+16 2 𝑥= 8 −16 2 𝑥= −8 2 =4 1 𝑥= 24 2 =12

So a = = 6 b = = 9 c = = -18 𝑥= − ± 2 −4 2 x= − 9 ± 9 2 −4 6 −18 2 6 𝑥= −9 ± 81+432 12 𝑥= −9 ± 513 12

Concept 7: evaluating and finding zeros of rational expressions.

Evaluating Rational Expressions Evaluating—Substituting a specific value in for x into a rational expressions and evaluating to find a specific answer. Evaluate 5 −𝑥 2𝑥+4 when x = 1 5 −(1) 2(1)+4 = 4 2+4 = 4 6 = 2 3

5 −(3) 2(3)+4 = 2 6+4 = 2 10 = 1 5 5 −(−2) 2(−2)+4 = 7 −4+4 = 7 0 = undefined

Evaluate 𝑥 2 −16 2𝑥+7 at the given values of x. a. x = 1 b. x = -5 c. x = 4 𝑥 2 −16 2𝑥+7 = (1) 2 −16 2(1)+7 = (−5) 2 −16 2(−5)+7 = (4) 2 −16 2(4)+7 = 1−16 2+7 = 25−16 −10+7 = 16−16 8+7 = −15 9 = 9 −3 = 0 15 = −5 3 or −1 2 3 =−3 =0

Finding zeros— finding the values of the variable that makes the denominator of a rational expression zero. Find the values that make the denominator zero.

Finding zeros— finding the values of the variable that makes the denominator of a rational expression zero.

Concept 8: Multiplying Rational Expressions

Multiplying Rational Numbers Step 1: Factor both the _________________ and the _______________ Step 2: Write together as one fraction, by ____________ _______. Step 3 : ___________ the expression. Step 4: Multiply _________ remaining factors in the _______________ or the ____________________. numerator denominator multiplying across Simplify any numerator denominator

1 1. = 12 𝑥 2 𝑥 3 24𝑦 = 1 𝑥 5 2𝑦 2 = 2 𝑎 2 𝑏 2 𝑎 𝑏 2 𝑐 = 2𝑎 𝑐 2.

= 4 3𝑥 = 2 𝑥−6 ∗14 7𝑥∗3(𝑥−6) 3. = 𝑥 3𝑥+1 6(𝑥−5) = 3𝑥 2 +𝑥 6𝑥−30 4. 2 2(x – 6) 2 = 2 𝑥−6 ∗14 7𝑥∗3(𝑥−6) = 4 3𝑥 3. 1 3(x – 6) = 𝑥 3𝑥+1 6(𝑥−5) = 3𝑥 2 +𝑥 6𝑥−30 4.

= 16 𝑥 2 𝑥−2 (𝑥+5) 8 𝑥 6 (𝑥+5) 5. = 2 𝑥−2 𝑥 4 = 2𝑥−4 𝑥 4 (x - 2)(x + 5) 2 = 16 𝑥 2 𝑥−2 (𝑥+5) 8 𝑥 6 (𝑥+5) 5. 1 𝑥 4 = 2 𝑥−2 𝑥 4 = 2𝑥−4 𝑥 4 5(x + 9) x = 15 𝑥 3 𝑥+9 𝑥 2 (𝑥+9)(𝑥 −9) = 15𝑥 𝑥−9 6. (x + 9)(x - 9)

= 𝑥+2 𝑥 2 +𝑥−6 = 𝑥+2 (𝑥−2)(𝑥+3) 7. 2𝑥 2 −6𝑥−56 𝑥−3 ∙ 𝑥 2 −9 𝑥 2 +5𝑥+6 (x + 2)(x - 6) = 𝑥+2 𝑥 2 +𝑥−6 7. = 𝑥+2 (𝑥−2)(𝑥+3) (x - 2)(x + 2) (x - 3)(x - 6) 2(𝑥−7)(𝑥+4) 2( 𝑥 2 −3𝑥−28) (𝑥−3)(𝑥+3) 2𝑥 2 −6𝑥−56 𝑥−3 ∙ 𝑥 2 −9 𝑥 2 +5𝑥+6 = 2(𝑥−7)(𝑥+4) (𝑥+2) 8. (𝑥+2)(𝑥+3) = 2( 𝑥 2 −10𝑥+4𝑥−28) 𝑥+2 = 2 𝑥 2 −12𝑥−56 𝑥+2

(x - 7)(x + 7) (x - 7)(x - 3) = (𝑥−7)(𝑥−7) 𝑥(𝑥+4)(𝑥+3) 9. 𝑥 2 −49 𝑥 3 +11 𝑥 2 +28𝑥 ∙ 𝑥 2 −10𝑥+21 𝑥 2 −9 (x - 3)(x + 3) 𝑥( 𝑥 2 +11𝑥+28) = 𝑥 2 −14𝑥+49 𝑥( 𝑥 2 +7𝑥+12) x(x + 4)(x + 7) = 𝑥 2 −14𝑥+49 𝑥 3 +7 𝑥 2 +12𝑥

= 𝑥(𝑥+11) 4(𝑥−2) 10. = 𝑥 2 +11𝑥 4𝑥−8 = 5𝑥(𝑥+7) 𝑥+5 11. = 5𝑥 2 +35𝑥 𝑥+5 (x + 11)(x + 11) = 𝑥(𝑥+11) 4(𝑥−2) 10. 4 2x(x - 2) = 𝑥 2 +11𝑥 4𝑥−8 5x (x + 7)(x - 7) = 5𝑥(𝑥+7) 𝑥+5 11. 1 = 5𝑥 2 +35𝑥 𝑥+5 2( 𝑥 2 −2𝑥−35) 2(𝑥−7)(𝑥+5)

Concept 9: Dividing Rational Expressions We ___________ divide rational expressions we ______________ by the ________________ of the ___________ expression. do not multiply reciprocal last

1. = 𝑟 3 𝑠 𝑡 ∙ 𝑡 3 𝑟 𝑠 3 = 𝑟 2 𝑡 2 𝑠 2 = 𝑢 5 𝑥 𝑦 ∙ 𝑦 𝑢 𝑥 2 = 𝑢 4 𝑥 2. = 𝑟 3 𝑠 𝑡 ∙ 𝑡 3 𝑟 𝑠 3 = 𝑟 2 𝑡 2 𝑠 2 𝑠 2 1 1 𝑢 4 1 1 = 𝑢 5 𝑥 𝑦 ∙ 𝑦 𝑢 𝑥 2 = 𝑢 4 𝑥 2. x 1 1

3. = 𝑚 5 𝑛 𝑝 ∙ 𝑝 4 𝑚 𝑛 4 = 𝑚 4 𝑝 3 𝑛 3 = 30 𝑦 2 +4𝑦−12 ∙ 𝑦−2 6𝑦 = 𝑚 5 𝑛 𝑝 ∙ 𝑝 4 𝑚 𝑛 4 = 𝑚 4 𝑝 3 𝑛 3 1 𝑛 3 1 5 1 = 30 𝑦 2 +4𝑦−12 ∙ 𝑦−2 6𝑦 = 5 𝑦+6 4. 1 𝑦−2 (𝑦+6) 1

5. 3 1 = 15 𝑦 2 +2𝑦−8 ∙ 𝑦−2 5𝑦 = 3 𝑦(𝑦+4) 𝑦 (𝑦−2)(𝑦+4) = 3 𝑦 2 +4𝑦 6. (𝑦+7)(𝑦−4) (𝑦+2)(𝑦−7) = 𝑥 2 +3𝑥−28 𝑥 2 +4𝑥+4 ∙ 𝑥 2 −5𝑥−14 𝑥 2 −49 (𝑦+2)(𝑦+2) (𝑦+7)(𝑦−7) = 𝑦−4 𝑦+2

7. = 𝑦 2 −9 𝑦 2 ∙ 𝑦+2 𝑦 5 +3 𝑦 4 = (𝑦−3)(𝑦−2) 𝑦 6 8. (𝑦+3)(𝑦−3) = 𝑦 2 −9 𝑦 2 ∙ 𝑦+2 𝑦 5 +3 𝑦 4 = (𝑦−3)(𝑦−2) 𝑦 6 𝑦 4 (𝑦+3) 4( 𝑥 2 −16) 8. x(𝑥−4) = 4𝑥 2 −64 8𝑥 ∙ 𝑥 2 −4𝑥 𝑥 2 −16 = 4(𝑥−4) 8 = 𝑥−4 2 2