Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { perform the indicated operations and simplify. } $$ $$ (4 x-11)(3 x-7) $$

Step-by-Step Solution

Answer

The simplified expression is \(12x^2 - 61x + 77\).

Step 1: Apply the Distributive Property

To solve \[(4x - 11)(3x - 7),\]we begin by applying the distributive property (also known as the FOIL method for binomials). Each term in the first binomial is multiplied by each term in the second binomial. This involves four multiplication operations:1. First: Multiply the first terms in each binomial, \((4x)(3x) = 12x^2\)2. Outer: Multiply the outer terms, \((4x)(-7) = -28x\)3. Inner: Multiply the inner terms, \((-11)(3x) = -33x\)4. Last: Multiply the last terms, \((-11)(-7) = 77\)

Step 2: Combine Like Terms

From Step 1, we obtained the terms:\[12x^2, -28x, -33x, \text{ and } 77.\]Next, we combine the like terms, which are the terms with \(x\):\(-28x - 33x = -61x.\)Now, write down the simplified expression combining all terms:\[12x^2 - 61x + 77.\]

Step 3: Write the Final Simplified Expression

The solution involves writing the fully simplified polynomial: \[12x^2 - 61x + 77.\] There are no further simplifications possible since all terms are combined and the polynomial is in standard form.

Key Concepts

Distributive Property

When working with polynomials, the distributive property is crucial. It allows you to multiply a single term by a group of terms inside parentheses. Here, we apply it to polynomials by multiplying each term in the first parentheses by each term in the second.

In this exercise, we have \[(4x - 11)(3x - 7)\]. Using the distributive property, we multiply:

• the first terms of each binomial, \((4x)\) and \((3x)\), resulting in the term \(12x^2\),
• the outer terms, \((4x)\) and \((-7)\), which gives \(-28x\),
• the inner terms, \((-11)\) and \((3x)\), yielding \(-33x\), and
• the last terms, \((-11)\) and \((-7)\), producing \(77\).

By following these steps, we ensure that each term from one binomial is multiplied with every term of the other, which is the essence of the distributive property. This results in a complete and expanded form of the polynomial.

FOIL Method

The FOIL method is a handy shortcut for applying the distributive property specifically to binomials. "FOIL" stands for First, Outer, Inner, Last, indicating the pairs of terms you multiply.

• First: Multiply the first terms of each binomial \((4x)\) and \((3x)\) to get \(12x^2\).
• Outer: Multiply the outermost terms \((4x)\) and \((-7)\) to obtain \(-28x\).
• Inner: Multiply the inside terms \((-11)\) and \((3x)\) to get \(-33x\).
• Last: The last terms \((-11)\) and \((-7)\) multiply to \(77\).

By organizing the multiplications this way, you simplify the process, ensuring nothing is missed. Each step corresponds to visibility of each multiplication pair, making it easier to keep track, which helps especially when learning polynomial multiplication.

Combining Like Terms

After using the distributive property or FOIL method, you'll end up with several terms. To finalize the expression, combine like terms, or terms that have the same variables raised to the same powers.

In our case, after expansion, we have:\[12x^2, -28x, -33x, \text{ and } 77.\]Focus on the terms involving the same degree of x:

• Combine \(-28x\) and \(-33x\), which simplifies to \(-61x\).

The terms \(12x^2\) and \(77\) remain unchanged since there are no other terms like them. The simplified equation becomes\[12x^2 - 61x + 77.\]Combining like terms is the final step to neatly simplify the polynomial into its standard form, making it clearer and easier to interpret.