Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

#1) Simplify and factor: 8x – 4x – 6x + 6 8x-4x-6x+6 Since 8x and -4x are like terms, add -4x to 8x to get 4x. 4x-6x+6 Since 4x and -6x are like terms, add -6x to 4x to get -2x. -2x+6 #2) Factor by grouping: x(x-1) – 3(x-1) x(x-1)-3(x-1) Factor out the GCF of (x-1) from each term in the polynomial. (x-1)(x)+(x-1)(-3) Factor out the GCF of (x-1) from x(x-1)-3(x-1). (x-1)(x-3) #3) Solve for x in the following equation: x(x-1) – 3(x-1) = 0 Hint: Use the answer of problem #2 (simplified expression). x(x-1)-3(x-1)=0 Factor out the GCF of (x-1) from each term in the polynomial. (x-1)(x)+(x-1)(-3)=0 Factor out the GCF of (x-1) from x(x-1)-3(x-1). (x-1)(x-3)=0 If any individual factor on the left-hand side of the equation is equal to 0, the entire expression will be equal to 0. (x-1)=0_(x-3)=0 Set the first factor equal to 0 and solve. (x-1)=0 Remove the parentheses around the expression x-1. x-1=0 Since -1 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 1 to both sides. x=1 Set the next factor equal to 0 and solve. (x-3)=0 Remove the parentheses around the expression x-3. x-3=0 Since -3 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 3 to both sides. x=3 The final solution is all the values that make (x-1)(x-3)=0 true. x=1,3 #4) Factoring trinomial: x^2 – 5x + 4 x^(2)-5x+4 2)+bx+c, find two factors of c (4) that add up to b (-5). In this problem -1*-4=4 and -1-4=-5, so insert -1 as the right hand term of one factor and -4 as the right-hand term of the other factor. (x-1)(x-4) #5) Factoring trinomial: x^3 – 5x^2 + 6x x^(3)-5x^(2)+6x Factor out the GCF of x from each term in the polynomial. x(x^(2))+x(-5x)+x(6) Factor out the GCF of x from x^(3)-5x^(2)+6x. x(x^(2)-5x+6) x^(2)+bx+c, find two factors of c (6) that add up to b (-5). In this problem -2*-3=6 and -2-3=-5, so insert -2 as the right hand term of one factor and -3 as the right-hand term of the other factor. x(x-2)(x-3) #6) Factoring trinomial: x^3 – 5x^2 – 6x x^(3)-5x^(2)-6x Factor out the GCF of x from each term in the polynomial. x(x^(2))+x(-5x)+x(-6) Factor out the GCF of x from x^(3)-5x^(2)-6x. x(x^(2)-5x-6) x^(2)+bx+c, find two factors of c (-6) that add up to b (-5). In this problem 1*-6=-6 and 1-6=-5, so insert 1 as the right hand term of one factor and -6 as the right-hand term of the other factor. x(x+1)(x-6) #7) Subtract these two rational expressions: 2/(3x) – 4/5 (2)/(3x)-(4)/(5) To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is 15x. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions. (2)/(3x)*(5)/(5)-(4)/(5)*(3x)/(3x) Complete the multiplication to produce a denominator of 15x in each expression. (10)/(15x)-(12x)/(15x) Combine the numerators of all expressions that have common denominators. (10-12x)/(15x) Reorder the polynomial 10-12x alphabetically from left to right, starting with the highest order term. (-12x+10)/(15x) #8) Subtract these two rational expressions: (2x+3)/(x-1) – (x-1)/(x+1) (2x+3)/(x-1)-(x-1)/(x+1) Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of (x+1)(x-1). The ((2x+3))/((x-1)) expression needs to be multiplied by ((x+1))/((x+1)) to make the denominator (x+1)(x-1). The -((x-1))/((x+1)) expression needs to be multiplied by ((x-1))/((x-1)) to make the denominator (x+1)(x-1). (2x+3)/(x-1)*(x+1)/(x+1)-(x-1)/(x+1)*(x-1)/(x-1) Multiply the expression by a factor of 1 to create the least common denominator (LCD) of (x+1)(x-1). ((2x+3)(x+1))/((x+1)(x-1))-(x-1)/(x+1)*(x-1)/(x-1) Multiply the expression by a factor of 1 to create the least common denominator (LCD) of (x+1)(x-1). ((2x+3)(x+1))/((x+1)(x-1))-((x-1)(x-1))/((x+1)(x-1)) The numerators of expressions that have equal denominators can be combined. In this case, ((2x+3)(x+1))/((x+1)(x-1)) and -(((x-1)(x-1)))/((x+1)(x-1)) have the same denominator of (x+1)(x1), so the numerators can be combined. ((2x+3)(x+1)-((x-1)(x-1)))/((x+1)(x-1)) Simplify the numerator of the expression. (x^(2)+7x+2)/((x+1)(x-1)) #9) Subtract these two rational expressions: 2/(3x-1) – 1/(x+1) (2)/(3x-1)-(1)/(x+1) Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of (3x-1)(x+1). The (2)/((3x-1)) expression needs to be multiplied by ((x+1))/((x+1)) to make the denominator (3x-1)(x+1). The -(1)/((x+1)) expression needs to be multiplied by ((3x1))/((3x-1)) to make the denominator (3x-1)(x+1). (2)/(3x-1)*(x+1)/(x+1)-(1)/(x+1)*(3x-1)/(3x-1) Multiply the expression by a factor of 1 to create the least common denominator (LCD) of (3x1)(x+1). (2(x+1))/((3x-1)(x+1))-(1)/(x+1)*(3x-1)/(3x-1) Multiply the expression by a factor of 1 to create the least common denominator (LCD) of (3x1)(x+1). (2(x+1))/((3x-1)(x+1))-(1(3x-1))/((3x-1)(x+1)) Remove the parentheses around the expression 3x-1. (2(x+1))/((3x-1)(x+1))-(3x-1)/((3x-1)(x+1)) The numerators of expressions that have equal denominators can be combined. In this case, (2(x+1))/((3x-1)(x+1)) and -((3x-1))/((3x-1)(x+1)) have the same denominator of (3x-1)(x+1), so the numerators can be combined. (2(x+1)-(3x-1))/((3x-1)(x+1)) Simplify the numerator of the expression. (2x+2+-3x+1)/((3x-1)(x+1)) Combine all similar terms in the polynomial 2x+2-3x+1. (-x+3)/((3x-1)(x+1)) #10) Solve for x in the following equation: 2/(3x-1) – 1/(x+1) = 0 Hint: Use the answer of problem #9 (simplified expression). (2)/(3x-1)-(1)/(x+1)=0 Find the LCD (least common denominator) of (2)/((3x-1))-(1)/((x+1))+0. Least common denominator: (x+1)(3x-1) Multiply each term in the equation by (x+1)(3x-1) in order to remove all the denominators from the equation. (2)/(3x-1)*(x+1)(3x-1)-(1)/(x+1)*(x+1)(3x-1)=0*(x+1)(3x-1) Simplify the left-hand side of the equation by canceling the common factors. -x+3=0*(x+1)(3x-1) Simplify the right-hand side of the equation by multiplying out all the terms. -x+3=0 Since 3 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 3 from both sides. -x=-3 Multiply each term in the equation by -1. -x*-1=-3*-1 Multiply -x by -1 to get x. x=-3*-1 Multiply -3 by -1 to get 3. x=3