Problem 102 - Algebra and Trigonometry

Question

Factor and simplify each algebraic expression. $$-8(4 x+3)^{-2}+10(5 x+1)(4 x+3)^{-1}$$

Step-by-Step Solution

Answer

The simplified and factored form of the expression is \[ \frac{50x+2}{4(2x+1.5)^2} \]

Step 1: Distribute the Negative Exponents

Rewrite the expression with negative exponents as fractions. The expression then becomes \[ -8 \div (4x+3)^2 + 10\cdot(5x+1) \div (4x+3) \]

Step 2: Isolate the Common Factors

Factorize the expressions under the reciprocal to create a more navigable expression: \[ -8 \div [(2(2x+1.5))^2] + 10\cdot(5x+1) \div (2(2x+1.5)) \]

Step 3: Further Simplify the Expression

Cancel out the twos in both denominators, and simplify the common factors in the denominators: \[ -8 \div (4(2x+1.5)^2) + 10\cdot(5x+1) \div (2(2x+1.5)) \]

Step 4: Combine the Expressions

After simplifying the denominators, combine the two fractions into one so that they have a common denominator. The expression now looks like this: \[ \frac{-8+10(5x+1)}{(4(2x+1.5)^2)} \]

Step 5: Simplify the Numerator

Simplify the numerator: \[ \frac{-8+50x+10}{4(2x+1.5)^2} \]

Step 6: Final Simplification

Carry out the addition in the numerator to produce the final simplified expression: \[ \frac{50x+2}{4(2x+1.5)^2} \]