Problem 61 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$\left(x^{2}+2 x+1\right)\left(x^{2}+3 x+4\right)$$

Step-by-Step Solution

Answer

The product is \(x^4 + 5x^3 + 11x^2 + 11x + 4\).

Step 1: Apply the Distributive Property

To multiply two polynomials, each term in the first polynomial should be multiplied by each term in the second polynomial. Start by distributing each term of the first polynomial, \(x^2 + 2x + 1\), to each term in the second polynomial, \(x^2 + 3x + 4\).

Step 2: Multiply the Quadratic Terms

First, distribute \(x^2\) from the first polynomial across all terms in the second polynomial. This results in: \[x^2(x^2 + 3x + 4) = x^4 + 3x^3 + 4x^2\]

Step 3: Multiply the Linear Terms

Next, distribute \(2x\) from the first polynomial across all terms in the second polynomial. Calculate as follows: \[2x(x^2 + 3x + 4) = 2x^3 + 6x^2 + 8x\]

Step 4: Multiply the Constant Term

Finally, distribute \(1\) from the first polynomial across all terms in the second polynomial, obtaining: \[1(x^2 + 3x + 4) = x^2 + 3x + 4\]

Step 5: Combine Like Terms

Now, combine all the terms obtained from the previous steps: \[x^4 + 3x^3 + 4x^2 + 2x^3 + 6x^2 + 8x + x^2 + 3x + 4\] Group and combine the like terms together as follows: 1. \(x^4\) 2. Combine \(3x^3\) and \(2x^3\) to get \(5x^3\). 3. Combine \(4x^2\), \(6x^2\), and \(x^2\) to get \(11x^2\). 4. Combine \(8x\) and \(3x\) to get \(11x\).5. The constant is \(4\).

Step 6: Write the Final Expression

After combining the like terms, the final product of the two binomials is:\[x^4 + 5x^3 + 11x^2 + 11x + 4\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms within a polynomial. It works by spreading, or distributing, the term across each addend inside parentheses. This can be expressed as:

• If you have \(a(b + c)\), then you distribute "\(a\)" to both "\(b\)" and "\(c\)," resulting in \(ab + ac\).
• In our exercise, we apply the distributive property for each term in the polynomial \((x^2 + 2x + 1)\) across each term in the polynomial \((x^2 + 3x + 4)\).

Using the distributive property correctly is key in polynomial multiplication. This property helps in breaking down complex expressions into simpler ones, making it easier to handle multiple terms.

Combining Like Terms

Combining like terms is an important step to simplify expressions. It involves grouping and summing terms with the same variable and degree.

• In an expression like \(3x^2 + 5x^2\), you can combine these terms because they both involve \(x^2\), resulting in \(8x^2\).
• In the final step of our exercise, after distributing all terms, we end up with several similar terms, such as \(x^4\), \(3x^3\), and \(2x^3\).

By combining like terms:

• Group all \(x^3\) terms which gives \((3x^3 + 2x^3 = 5x^3)\).
• Do the same for \(x^2\) terms \((4x^2 + 6x^2 + 1x^2 = 11x^2)\).

This simplifies the polynomial into a more compact form by reducing the number of terms and making it easier to analyze or solve.

Binomials

A binomial is a polynomial consisting of exactly two terms. Each term is typically separated by a plus or minus sign. Binomials play a crucial role in simplifying algebraic equations and often appear in polynomial multiplication.

• Examples of binomials include \(x + 3\), \(y - 7\), or \(2x^2 + 5x\).
• In the exercise, the expression \((x^2 + 2x + 1)(x^2 + 3x + 4)\) considers expanded versions of binomials within itself.

When multiplying binomials, we use the distributive property, often expanding it into a process known as "FOIL" for first, outer, inner, and last terms. Despite having more than two terms in this case, understanding binomial multiplication basics is essential to build up to multiplying larger polynomials.

Quadratic Terms

Quadratic terms are terms of a polynomial where the variable is raised to the power of two. These terms are significant components in quadratic equations, which form the basis of many algebraic applications.

• An example is \(3x^2\), where 2 is the highest degree of the variable \(x\).
• In our problem, several quadratic terms appear from distributing and multiplying, including \(4x^2, 6x^2,\) and \(x^2\).

In solving polynomial multiplication, identifying and combining these like quadratic terms simplifies the expression and makes it easier to interpret. Quadratic terms also serve as a critical step in understanding how individual components of an expression combine during multiplication.