simpliy the following: (a) {[4x^2-49]/[8x^3+27]} *{ [4x^2+12x+9]/[2x^2-13x+21]} (b) {[x^2+x-6]/[x^2+7x-6]}*{[20x^2-7x-3]/[6x^2-25x+4]}*{[4x^2-11x-3]/[6x^2-19x+3]}
Here's (a):
{[4x^2-49]/[8x^3+27]} *{ [4x^2+12x+9]/[2x^2-13x+21]}
4x^2-49 = (2x-7)(2x+7)
8x^3+27 = (2x+3)(4x^2-6x+9)
4x^2+12x+9 = (2x+3)(2x+3)
2x^2-13x+21 = (2x-7)(x-3)
Putting it all together, you can cancel the (2x-7)(2x+3) to get
(2x+7)(2x+3) / (x-3)(4x^2-6x+9)
Do (b) likewise. The key is to know how to factor these babies. Recognize differences of square and cubes.
simpliy the expression by adding like terms and write the expression in descending powers of x 4x²+11x-3x²-12+x
To simplify the given expressions, we need to first factorize the numerator and denominator of each fraction and then cancel out common factors. Let's simplify each expression step by step.
(a)
First, let's factorize each fraction:
1. Factorize the numerator of the first fraction [4x^2-49]:
[4x^2-49] = (2x+7)(2x-7)
2. Factorize the denominator of the first fraction [8x^3+27]:
[8x^3+27] = (2x+3)(4x^2-6x+9)
3. Factorize the numerator of the second fraction [4x^2+12x+9]:
[4x^2+12x+9] = (2x+3)(2x+3)
4. Factorize the denominator of the second fraction [2x^2-13x+21]:
[2x^2-13x+21] cannot be factored further.
Now, rewrite the expression with the factored forms:
{ [(2x+7)(2x-7)] / [(2x+3)(4x^2-6x+9)] } * { [(2x+3)(2x+3)] / [2x^2-13x+21] }
Next, we can start canceling out common factors between the numerators and denominators:
{ [(2x+7)(2x-7)] / [(2x+3)(2x+3)(2x^2-13x+21)] } * { 1 / [2x^2-13x+21] }
Finally, we can cancel out the remaining common factor between the numerators and denominators:
(2x+7)(2x-7)
So, the simplified expression is (2x+7)(2x-7).
(b)
Follow the same steps as in part (a) to factorize and simplify each fraction. Let's go step by step:
1. Factorize the numerator of the first fraction [x^2+x-6]:
[x^2+x-6] = (x+3)(x-2)
2. Factorize the denominator of the first fraction [x^2+7x-6]:
[x^2+7x-6] = (x+6)(x-1)
3. Factorize the numerator of the second fraction [20x^2-7x-3]:
[20x^2-7x-3] = (5x-1)(4x+3)
4. Factorize the denominator of the second fraction [6x^2-25x+4]:
[6x^2-25x+4] = (2x-1)(3x-4)
5. Factorize the numerator of the third fraction [4x^2-11x-3]:
[4x^2-11x-3] = (4x+1)(x-3)
6. Factorize the denominator of the third fraction [6x^2-19x+3]:
[6x^2-19x+3] = (2x-1)(3x-3)
Now, rewrite the expression with the factored forms:
{ [(x+3)(x-2)] / [(x+6)(x-1)] } * { [(5x-1)(4x+3)] / [(2x-1)(3x-4)] } * { [(4x+1)(x-3)] / [(2x-1)(3x-3)] }
Next, we can start canceling out common factors between the numerators and denominators:
{ [(x+3)(x-2)] / [(x+6)(x-1)] } * { [(5x-1)(4x+3)(4x+1)(x-3)] / [(2x-1)(2x-1)(3x-4)(3x-3)] }
Finally, we can cancel out the remaining common factors:
(x+3)(5x-1)(4x+1)(x-3) / (x-1)
So, the simplified expression is (x+3)(5x-1)(4x+1)(x-3) / (x-1).