Problem 86 - College Algebra Essentials

Question

Perform the indicated operation or operations. $$(3 x+5)(2 x-9)-(7 x-2)(x-1)$$

Step-by-Step Solution

Answer

The simplified expression after performing the stated operations is: \( -x^2 -8x -47 \)

Step 1: Multiply the two pairs of binomial

We start by multiplying the binomials \((3x+5)(2x-9)\) and \((7x-2)(x-1)\) separately. Remember, to multiply binomials, we use a method called FOIL. It stands for First, Outer, Inner and Last which represent how to multiply the terms of the binomials. For \((3x+5)(2x-9)\), it is calculated as follows: \[ (3x \cdot 2x) + (3x \cdot -9) + (5 \cdot 2x) + (5 \cdot -9) \] . Apply the similar method for the second pair of binomial.

Step 2: Simplify the Multiplication

Next, we simplify the multiplication, which gives: \[ (6x^2 -27x + 10x -45) - (7x^2 -2x -7x +2) \]

Step 3: Combine Like Terms

This step involves combining similar terms in the polynomials. This would result in: \[ (6x^2 -17x -45) - (7x^2 -9x +2) \] .

Step 4: Subtract Polynomials

To subtract the polynomials, we distribute the negative sign across the second polynomial and then combine like terms: \[ 6x^2 -17x -45 -7x^2 +9x -2 \] . This simplifies to \[ -x^2 -8x -47 \] .