Problem 86 - Algebra and Trigonometry

Question

In Exercises 83–90, perform the indicated operation or operations.. $$ (3 x+5)(2 x-9)-(7 x-2)(x-1) $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-x^2 - 8x - 47\).

Step 1: Apply distributive property on the first polynomial

Multiply the first two binomials. Multiply \(3x\) by \(2x\) and by \(-9\), and then do the same for \(5\), which results in: \(6x^2 - 27x + 10x - 45\). This simplifies to \(6x^2 - 17x - 45\).

Step 2: Apply distributive property on the second polynomial

Multiply the next two binomials next. Multiply \(7x\) by \(x\) and by \(-1\), and then do the same for \(-2\), which results in: \(7x^2 - 7x - 2x + 2\). This simplifies to \(7x^2 - 9x + 2\).

Step 3: Subtract the second polynomial from the first

Now that we have the simplified forms of both polynomials, subtract the second one from the first. We get \( (6x^2 - 17x - 45) - (7x^2 - 9x + 2)\). It’s important to note that every term of the second polynomial changes sign because of the subtraction. So we get \(6x^2 - 17x - 45 - 7x^2 + 9x - 2\). Arrange like terms together we get \(-x^2 - 8x - 47\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in mathematics. It allows us to simplify expressions and solve equations effectively. By using the distributive property, you can multiply a single term by two or more terms inside parentheses.

In mathematical terms, if you have an expression like \(a(b + c)\), the property lets you "distribute" \(a\) across each term inside the parentheses, resulting in: \[a(b + c) = ab + ac\]In the exercise, the distributive property was used twice:

• First, it was applied to the binomials \((3x + 5)\) and \((2x - 9)\), resulting in the expanded expression \(6x^2 - 27x + 10x - 45\).
• Second, the property was applied to the binomials \((7x - 2)\) and \((x - 1)\), resulting in \(7x^2 - 7x - 2x + 2\).

This crucial step prepares the expressions for further operations, like subtraction or addition.

Binomial Multiplication

Binomial multiplication is when you multiply two binomials, each with two terms. It involves using the distributive property to ensure every term in the first binomial is multiplied by each term in the second binomial.

The FOIL method is a handy way to remember how to do this. FOIL stands for:

• First terms multiplied together.
• Outer terms multiplied.
• Inner terms multiplied.
• Last terms multiplied together.

For example, in our problem, when multiplying \((3x + 5)\) by \((2x - 9)\), the FOIL method results in:\[3x(2x) + 3x(-9) + 5(2x) + 5(-9)\]which simplifies down to \(6x^2 - 27x + 10x - 45\). Following this method ensures that all parts of the polynomials are accounted for in the multiplication.

Simplifying Expressions

Simplifying expressions is the process of reducing an expression to its simplest form. This often involves combining like terms, which are terms that have the same variable raised to the same power.

In the exercise solution, both expanded binomials were first simplified:

• The expression \(6x^2 - 27x + 10x - 45\) was simplified to \(6x^2 - 17x - 45\).
• The expression \(7x^2 - 7x - 2x + 2\) became \(7x^2 - 9x + 2\).

After this simplification, the second polynomial was then subtracted from the first:\[(6x^2 - 17x - 45) - (7x^2 - 9x + 2)\]This involves changing the sign of each term in the second polynomial before combining it with the first. The resulting expression, after combining like terms, is \(-x^2 - 8x - 47\). Simplifying is essential for making expressions easier to work with and solve.