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Chapter 4: Problem 14

Find the following products and simplify. $$ (a+1)(a+4) $$

Short Answer

Expert verified
Answer: The product of (a + 1) and (a + 4) when simplified is \(a^2 + 5a + 4\).

Step by step solution

01

Apply the FOIL method or the distributive property

We need to multiply each term of the first binomial with every term from the other binomial. Using the FOIL method: - First terms: \(a * a = a^2\) - Outer terms: \(a * 4 = 4a\) - Inner terms: \(1 * a = a\) - Last terms: \(1 * 4 = 4\) Now, we can write the product as: $$ a^2 + 4a + a + 4 $$
02

Simplify the expression

Combine like terms: $$ a^2 + (4a + a) + 4 $$ This simplifies to: $$ a^2 + 5a + 4 $$ So, the final product after simplification is: $$ (a+1)(a+4)=a^2+5a+4 $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

FOIL Method
The FOIL Method is a technique for expanding the product of two binomials. FOIL stands for First, Outer, Inner, Last, which are the positions of the pair of terms to be multiplied. Here's how it works:

  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in the product.
  • Inner: Multiply the inner terms.
  • Last: Multiply the last terms.
After performing these operations, you are left with four terms that may contain like terms, which should be combined to simplify the expression. For example, using the FOIL Method for \(a+1)(a+4)\) gives us \(a^2\), \(4a\), \(a\), and \(4\), which we then combine to \(a^2 + 5a + 4\).
Distributive Property
The distributive property is a fundamental algebraic property used in arithmetic and algebra. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. The formal expression of the property is \(a(b+c) = ab + ac\).

When it comes to polynomial expressions, the distributive property allows you to multiply each term of the first polynomial by each term of the second polynomial. In the example \(a+1)(a+4)\), we distribute each term of the first binomial through the second binomial, resulting in a series of simpler multiplications that we combine later.
Combining Like Terms
Combining like terms is an essential step in simplifying algebraic expressions and solving equations. Like terms are terms that contain the same variables raised to the same power. For example, \(4a\) and \(a\) are like terms because they both contain the variable \(a\) to the 1st power.

To combine like terms, you simply add or subtract the coefficients of these terms. Continuing with our example, combining the like terms \(4a\) and \(a\) from the expression \(a^2 + 4a + a + 4\) results in a new term, \(5a\), leading to the simplified expression \(a^2 + 5a + 4\).
Binomial Multiplication
Binomial multiplication refers to the process of multiplying two binomials together. Each term in the first binomial must be multiplied by each term in the second binomial, a task for which both the FOIL method and the distributive property are commonly used.

In practical terms, when we multiply \(a+1)\) by \(a+4)\), we are performing binomial multiplication. This process generates four terms, which are then simplified. The resulting product of two binomials is often a quadratic polynomial, as seen with the example that simplifies to \(a^2 + 5a + 4\), which is a standard form quadratic expression.

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Most popular questions from this chapter

Find the domain of the equations. Assume that the independent variable is the variable that appears in the expression to the right of the equal sign. $$ y=\frac{2}{x} $$

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