Problem 47 - Algebra A Combined Function

Question

Multiply. \((x+5)\left(x^{3}-3 x+4\right)\)

Step-by-Step Solution

Answer

The expression expands to \(x^4 + 5x^3 - 3x^2 - 11x + 20\).

Step 1: Distribute the first term of the first polynomial

Take the first term of \((x+5)\), which is \(x\), and multiply it with each term in \(x^3 - 3x + 4\).\[ x(x^3 - 3x + 4) = x imes x^3 + x imes (-3x) + x imes 4 \] This simplifies to: \[ x^4 - 3x^2 + 4x \]

Step 2: Distribute the second term of the first polynomial

Take the second term of \((x+5)\), which is \(5\), and multiply it with each term in \(x^3 - 3x + 4\).\[ 5(x^3 - 3x + 4) = 5 imes x^3 + 5 imes (-3x) + 5 imes 4 \] This simplifies to: \[ 5x^3 - 15x + 20 \]

Step 3: Combine like terms

Add the results from Step 1 and Step 2. Combine any like terms from \(x^4 - 3x^2 + 4x + 5x^3 - 15x + 20\).\[ x^4 + 5x^3 - 3x^2 + (4x - 15x) + 20 \] Simplifies to: \[ x^4 + 5x^3 - 3x^2 - 11x + 20 \]

Key Concepts

distributive property

The distributive property is a fundamental algebraic principle. It allows us to simplify expressions by distributing multiplication over addition or subtraction.
For polynomials, it involves multiplying each term of one polynomial by each term of another. This property is very useful for multiplication and combining expressions.
In the exercise given, we start by taking each term in the first polynomial,

• First term: \(x\)
• Second term: \(5\)

Each of these terms is multiplied by every term in the polynomial \(x^3 - 3x + 4\). By applying the distributive property:

• \(x \times (x^3 - 3x + 4)\) results in \(x^4 - 3x^2 + 4x\)
• \(5 \times (x^3 - 3x + 4)\) results in \(5x^3 - 15x + 20\)

Understanding this step-by-step application ensures you can distribute terms correctly in any algebraic equation. This property also sets the stage for effectively combining like terms later.

combine like terms

Combining like terms is the next crucial step in simplifying polynomial expressions. A 'like term' comes with variables raised to the same power. Identifying these terms allows us to simplify the expression further.
In our example, after using the distributive property, we obtain the expression:

• \(x^4 - 3x^2 + 4x + 5x^3 - 15x + 20\)

Notice the terms that share the same variables:

• Constant terms: \(20\)
• \(x\) terms: \(4x\) and \(-15x\)
• \(x^3\) terms: \(5x^3\)
• \(x^2\) terms: \(-3x^2\)
• \(x^4\) terms: \(x^4\)

By combining like terms \(4x - 15x\), we simplify these to \(-11x\). Thus, our final simplified expression becomes \(x^4 + 5x^3 - 3x^2 - 11x + 20\). Combining like terms effectively reduces the expression, making it more manageable and easier to work with.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations: addition, subtraction, multiplication, and sometimes division. They are fundamental in algebra as they represent mathematical relationships in a simplified form.
Polynomials are a type of algebraic expression and consist of terms that are numbers or products of numbers and variables raised to whole number powers.

• In the given example, \((x+5)\) and \((x^3 - 3x + 4)\) are both algebraic expressions.
• The expression \(x^4 + 5x^3 - 3x^2 - 11x + 20\) is the result of multiplying and simplifying these two polynomials.

The structure of algebraic expressions allows us to perform operations like multiplication through the distributive property and to simplify them by combining like terms.Understanding how to handle algebraic expressions is crucial, as they form the basis for solving equations and performing algebraic computations. They help in modeling real-world scenarios in mathematics, enabling deeper analytical thinking and problem-solving skills.