The document provides examples of common mistakes made when working with exponents, radicals, polynomials, rational expressions, and solving equations in pre-calculus. It demonstrates incorrect work and explains the errors, such as adding instead of multiplying terms with exponents or failing to square both sides of an equation when taking the square root of each side. Key concepts are reviewed, such as factoring polynomials before simplifying rational expressions or dividing terms. Helpful formulas are also listed, such as the quadratic formula.

1. Right and Wrong’s of Pre-calculus By: Njeri Johnson

2. Exponents(Rules) Any number (except 0) raised to the zero power is equal to 1:490= 1 Any number raised to the first power is always equal to itself. 81 = 8 If a number is raised to the second power, we say it is squared. 42is read as four squared.If a number is raised to the third power, we say it is cubed. 43is read as four cubed

3. Exponents Cont. 5x5x5=15 1/y⁻⁴ = -1/y⁴ ⁸√(x+5)⅛=(x+5)⁸ The first one is wrong because the numbers were added instead of multiplied together which is a common error is solving exponents. The second equation the numerator was made negative, but in reality the numerator and denominator would have to be switched thus making the negative exponent positive. The third one is wrong because instead of canceling the 8’s out the 1 was taken away.

4. Exponents Cont. 5x5x5=15 1/y⁻⁴ = -1/y⁴ ⁸√(x+5)⅛=(x+5)⁸ 5x5x5=125 1/y⁻⁴= y⁴/1 ⁸√(x+5⅛)=x+5

5. Radicals √-4=2 √9=3 √10=5 The first radical equation is wrong because the solution to the square root of a negative number is always a complex number, not a real number. Although the second radical equation is true, 3 is not the only solution for the square root of 9, the answer can also be -3. Many people take the number under a square root and divide it by two, but that’s not the correct way.

6. Radicals Cont. √-4=2 √9=3 √10=5 √-4=i√4 or complex number √9= +3 and -3 √10≈ 3.1622

7. Polynomials and Factoring Warning: Don't make the mistake of thinking that "factoring" means "dividing something off and making it magically disappear". Remember that "factoring" means "dividing out and putting in front of the parentheses". Nothing "disappears" when you factor; things are just arranged.

8. Polynomials and Factoring Cont. Factor In the second polynomial all the signs are positive in the question so the answer should be positive as well. The third polynomial was solved instead of factored. 2(x+3)=(x+3)(x+3) X2 +6+9=(x+3)(x-3) 3x – 12->3x=-12->x=-4 The first polynomial is actually factored to it’s simplest form already.

9. Polynomial and Factoring Cont. Factor 2(x+3)=(x+3)(x+3) X2 +6+9=(x+3)(x-3) 3x – 12->3x=-12 ->x=-4 2(x+3)= 2(x-3) x2+6+9=(x+3)(x+3) 3x-12=3(x-4)

10. Rational Expressions You can only cancel factors, not terms! When simplifying rational expressions cancel off any common numerical or variable factors. Simplifying rational expressions involves a lot of factoring.

11. Rational Expressions Cont. Simplify 2x2 +13x+20/2x2 +17x+30 = 13x +2/17x+3 X-2/2-x = 0 X+3 /X2 -X-12= X2 -4 The first rational expression was simplified by its terms and not its factors. The second rational expression was overlooked, 2-x is actually –x+2 which cant be simplified with x-2. The third rational expression was simplified by its terms and not its factors, the polynomial would have to be factored first.

12. Rational Expressions Cont. 2x2 +13x+20/2x2 +17x+30 = 13x +2/17x+3 X-2/2-x = 0 X+3 /X2 -X-12= X2 -4 2x2 +13x+20/2x2 +17x+30=(2x+5)(x+4)/(2x+5)(x+6)=(x+4)/(x+6) X-2/2-X=X-2/2-X X+3/X2-X-12=X+3/(X+3)(X-4)=X-4

13. Solving Equations Linear equations are just equations with a plain old variable like ‘X’ rather something complicated like ‘X²’ or ‘X/Y’. In general solving an equation involves undoing whatever has been done to the variable. Which is basically putting the variable by itself. Whatever you do to an equation,do the S A M E thingto B O T H sides of that equation!

14. Solving Equations Cont. Solve X-4=10->X=6 √X-8=3 ->(√X-8)²=3->X-8=3->x=11 X²-6X+9=(X-3)(X-3) The first equation is wrong because 4 was subtracted from 10 instead of added. In the second equation both sides weren’t squared. In the third equation the polynomial was only factored not solved.

15. Solving Equations Cont. Solve X-4=10->X=6 √X-8=3 ->(√X-8)²=3 ->X-8=3 ->x=11 X²-6X+9=(X-3)(X-3) X-4=10->X=14 √X-8=3->(√X-8)²=(3)²->X-8=9->X=17 X²-6X+9=(X-3)(X-3) ->X-3=0 X-3=0 X₁,₂ = 3

16. Helpful Theorems/Formulas -b⁺₋√b²-4ac/2a-> Quadratic formula (a-b)(a+b)=a²-b² (x+y)²=x²+2xy+y² Steps In Solving Radical Equations - Isolate the Radical - Square or cube both sides(depends on index) - Solve like a regular quadratic or linear equation