Multiplying and Dividing Rational Expressions:
2) x^2-8x+7/x^2+6x-7 = (x-7)/(x+7)
3) x^2-5x+6/x+4 times 3x+12/x-2 =3(x-3)
4)2x-3/5x+1 divided by 6x^2-13x+6/15x^2-7x-2=1
5) x^3-9x/x^2+11x+24 times x^2+7x-8/x^2-4x+3= x(x-3)(x-1)/(x+3)(x+8)
6) 4x-8/x^2-x-6 divided by x^3 +x^2-6x/x^2-9=(x-3)/x(x-2)
7) x^2-16/x-3 divided x+4/x^2-9=(x-1)/(x+1)
are they correct?
4. (2X-3)/(4X-1) -Div by (6X^2-13X+6)/
(15X^2-7X-2).
Factor trinomial in numerator using A*C Method:
A*C = 6*6 = 36 = -4*-9,
6X^2 -4X-9X +6
Group the 4 terms in pairs and factor each pair:
2X(3X-2) -3(3X-2). Factor out (3x-2):
(3X-2) (2X-3).
Factor the trinomial in denominator
using A*C Method:
A*C = 15*-2 =-30 = 3*-10.
15X^2 +3X-10X -2.
Group the 4 terms in pairs and factor each pair:
3x(SX+1) -3(5X+1), Factor out (5x+1):
(5X+1) (3X-2).
Write original expression in factored form:
(2X-3)/(5X+1) Div by (3X-2)(2X-3)/
(5X+1)(3X-2).
Cancel the (3X-2) factors:
(2X-3)/5X+1) Div by (2X-3)/(5X+1).
Invert divisor and multiply:
(2X-3)/(5X+1) * (5X+1)/(2X-3).
Cancel the (5X+1) factors:
(2X-3)/(2X-3) = 1.
5. (X^3-9)/(X^2+11X+24) * (X^2+7X-8_/
(X^2-4X+3) =
Factor both numerators & denominators:
X(X^2-9)/(X+3)(X+8) * (X-1)(X+8)/
(X-1)(X-3) =
Cancel factors (X+8), and (X-1).
X(X^2-9)/(X+3) * 1/(X-3)
Factor (X^2-9):
X(X+3)(X-3)/(X+3) * 1/(X-3).
Cancel factors (x+3) & (X-3):
= X.
6.(4X -8)/(X^2 -X -6) Div (X^3+X^26X)/
(X^2-9)
Factor completely:
4(X-2)/(X-3)(X+2) Div X(X^2+X-6)/(X+3)
(X-3).
4(X-2)/(X-3)(X+2) Div X(X+3)(X-2)/(X+3)(X-3).
Cancel factor (X+3) and invert divisor
4(X-2)/(X-3)(X+2) * (X-3)/X(X-2).
Cancel factors (X-2), (Z-3)
4/X(X+2).
To simplify rational expressions, we need to factor the numerators and denominators, cancel out any common factors, and then simplify further if possible. Let's go through each problem step by step:
2) x^2-8x+7/x^2+6x-7 = (x-7)/(x+7)
First, factor the numerator and denominator:
(x^2-8x+7) = (x-7)(x-1)
(x^2+6x-7) = (x+7)(x-1)
Next, cancel out the common factor of (x-1):
(x-7)/(x+7)
3) x^2-5x+6/x+4 times 3x+12/x-2 =3(x-3)
First, factor the numerator and denominator:
(x^2-5x+6) = (x-2)(x-3)
(3x+12) = 3(x+4)
Next, cancel out the common factor of (x+4):
(x-2)/(x-3) = 3(x-3)/(x-2)
You can also simplify it further by canceling out (x-2):
3(x-3)
4) 2x-3/5x+1 divided by 6x^2-13x+6/15x^2-7x-2 = 1
First, factor the numerators and denominators:
(2x-3)/(5x+1) = (2x-3)/(5x+1)
(6x^2-13x+6) = (2x-1)(3x-6)
(15x^2-7x-2) = (5x+2)(3x-1)
Next, rewrite the division as a multiplication by flipping the second fraction:
(2x-3)/(5x+1) * (3x-1)/(2x-1)
Now, cancel out the common factors:
(3x-1)/(5x+1)
5) x^3-9x/x^2+11x+24 times x^2+7x-8/x^2-4x+3 = x(x-3)(x-1)/(x+3)(x+8)
First, factor the numerators and denominators:
(x^3-9x) = x(x^2-9)
(x^2+11x+24) = (x+3)(x+8)
(x^2+7x-8) = (x+8)(x-1)
(x^2-4x+3) = (x-3)(x-1)
Next, cancel out the common factors:
x(x-3)(x-1)/(x+3)(x+8)
6) 4x-8/x^2-x-6 divided by x^3 + x^2 - 6x/x^2-9 = (x-3)/x(x-2)
First, factor the numerators and denominators:
(4x-8) = 4(x-2)
(x^2-x-6) = (x-3)(x+2)
(x^3 + x^2 - 6x) = x(x-3)(x+2)
(x^2-9) = (x-3)(x+3)
Next, rewrite the division as a multiplication by flipping the second fraction:
4(x-2)/(x-3)(x+2) * x(x-3)(x+2)/(x-3)(x+3)
Now, cancel out the common factors:
(x-2)/(x+3) = (x-3)/x(x-2)
7) x^2-16/x-3 divided x+4/x^2-9 = (x-1)/(x+1)
First, factor the numerators and denominators:
(x^2-16) = (x-4)(x+4)
(x-3) = (x-3)
(x+4) = (x+4)
(x^2-9) = (x-3)(x+3)
Next, rewrite the division as a multiplication by flipping the second fraction:
(x-4)(x+4)/(x-3) * (x+3)/(x+4)
Now, cancel out the common factors:
(x-4)/(x-3) = (x-1)/(x+1)
Remember to always check your work by multiplying the simplified expressions to make sure they are equal to the original expressions.