Break down the expression into basic terms: $$ 3x^2 + 2x $$

$$ 3\cdot x^2 + x + 2\cdot x $$

Algebraic Method Practice Problems with Solutions

Understanding Algebraic Technique

Complete explanation with examples

Algebraic Method

Algebraic Method is a general term for various tools and techniques that will help us solve more complex exercises in the future. It is mostly concern about using algebraic operations to isolate variables and solve equations. This approach is fundamental for solving equations in various mathematical contexts.

Distributive Property

This property helps us to clear parentheses and assists us with more complex calculations. Let's remember how it works. Generally, we will write it like this:

$Z\times(X+Y)=ZX+ZY$

$Z\times(X-Y)=ZX-ZY$

Extended Distributive Property

The extended distributive property is very similar to the distributive property, but it allows us to solve exercises with expressions in parentheses that are multiplied by other expressions in parentheses.
It looks like this:

$(a+b)\times(c+d)=ac+ad+bc+bd$

Factoring

The factoring method is very important. It will help us move from an expression with several terms to one that includes only one by taking out the common factor from within the parentheses.
For example:
$2A + 4B$

This expression consists of two terms. We can factor it by reducin by the greatest common factor. In this case, it's the $2$ .
We will write it as follows:

$2A+4B=2\times(A+2B)$

In this article, we’ll explain each of these topics in detail, But each of these topics will be explained even more in detail in their respective articles.

Detailed explanation

Practice Algebraic Technique

Test your knowledge with 58 quizzes

$$ (a+4)(c+3)= $$

Examples with solutions for Algebraic Technique

Step-by-step solutions included

Exercise #1

$(3+20)\times(12+4)=$

Step-by-Step Solution

Simplify this expression paying attention to the order of arithmetic operations. Exponentiation precedes multiplication whilst division precedes addition and subtraction. Parentheses precede all of the above.

Therefore, let's first start by simplifying the expressions within the parentheses. Then we can proceed to perform the multiplication between them:

$(3+20)\cdot(12+4)=\\ 23\cdot16=\\ 368$

Therefore, the correct answer is option A.

Answer:

368

Video Solution

Exercise #2

$(12+2)\times(3+5)=$

Step-by-Step Solution

Simplify this expression by paying attention to the order of arithmetic operations which states that exponentiation precedes multiplication, division precedes addition and subtraction and that parentheses precede all of the above.

Thus, let's begin by simplifying the expressions within the parentheses, and following this, the multiplication between them.

$(12+2)\cdot(3+5)= \\ 14\cdot8=\\ 112$

Therefore, the correct answer is option C.

Answer:

112

Video Solution

Exercise #3

$(35+4)\times(10+5)=$

Step-by-Step Solution

We begin by opening the parentheses using the extended distributive property to create a long addition exercise:

We then multiply the first term of the left parenthesis by the first term of the right parenthesis.

We multiply the first term of the left parenthesis by the second term of the right parenthesis.

Now we multiply the second term of the left parenthesis by the first term of the left parenthesis.

Finally, we multiply the second term of the left parenthesis by the second term of the right parenthesis.

In the following way:

$(35\times10)+(35\times5)+(4\times10)+(4\times5)=$

We solve each of the exercises within parentheses:

$350+175+40+20=$

We solve the exercise from left to right:

$350+175=525$

$525+40=565$

$565+20=585$

Answer:

585

Video Solution

Exercise #4

Break down the expression into basic terms:

$4x^2 + 6x$

Step-by-Step Solution

To break down the expression $4x^2 + 6x$ into its basic terms, we need to look for a common factor in both terms.

The first term is $4x^2$ , which can be rewritten as $4\cdot x\cdot x$ .

The second term is $6x$ , which can be rewritten as $2\cdot 3\cdot x$ .

The common factor between the terms is $x$ .

Thus, the expression can be broken down into $4\cdot x^2 + 6\cdot x$ , and further rewritten with common factors as $4\cdot x\cdot x + 6\cdot x$ .

Answer:

$4\cdot x\cdot x+6\cdot x$

Exercise #5

Simplify the expression:

$5x^3 + 3x^2$

Step-by-Step Solution

To simplify the expression $5x^3 + 3x^2$ , we can break it down into basic terms:

The term $5x^3$ can be written as $5 \cdot x \cdot x \cdot x$ .

The term $3x^2$ can be written as $3 \cdot x \cdot x$ .

Thus, the expression simplifies to $5 \cdot x \cdot x \cdot x + 3 \cdot x \cdot x$ .

Answer:

$5\cdot x\cdot x\cdot x + 3\cdot x\cdot x$

Frequently Asked Questions

Everything you need to know about Algebraic Technique

What is the distributive property in algebra?

+

The distributive property states that Z × (X + Y) = ZX + ZY. This means you multiply the term outside the parentheses by each term inside the parentheses. It's essential for clearing parentheses and simplifying algebraic expressions.

How do you factor algebraic expressions?

+

To factor an expression, find the greatest common factor (GCF) of all terms. For example, 2A + 4B = 2(A + 2B) because 2 is the GCF. Follow these steps: 1. Identify the GCF of all terms 2. Divide each term by the GCF 3. Write the GCF outside parentheses with remaining terms inside

What's the difference between (-4)² and -4²?

+

(-4)² = (-4) × (-4) = 16 because the negative sign is inside parentheses and gets squared. However, -4² = -(4 × 4) = -16 because you calculate the exponent first, then apply the negative sign.

How does the extended distributive property work?

+

The extended distributive property multiplies two binomials: (a+b) × (c+d) = ac + ad + bc + bd. Each term in the first parentheses multiplies each term in the second parentheses, creating four products that you then combine.

When should I use factoring vs distributive property?

+

Use the distributive property to expand expressions (remove parentheses). Use factoring to simplify expressions by pulling out common factors (create parentheses). They're opposite processes - choose based on whether you want to expand or simplify.

What are common mistakes with exponents in algebra?

+

Common exponent mistakes include: • Confusing (-a)ⁿ with -aⁿ • Forgetting that a⁰ = 1 for any non-zero number • Not following order of operations (parentheses before exponents) • Incorrectly distributing exponents over addition: (a+b)² ≠ a² + b²

How do algebraic methods help solve equations?

+

Algebraic methods like distributive property and factoring help isolate variables by simplifying expressions. They allow you to clear parentheses, combine like terms, and manipulate equations systematically to find solutions.

What grade level learns these algebraic techniques?

+

Students typically learn the distributive property around age 12 (6th-7th grade), factoring in 8th-9th grade, and extended distributive property in Algebra I (9th-10th grade). These form the foundation for advanced algebraic problem-solving.

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems

Applying the formula Ascertain whether the law of distributive property is applicable Complete the missing number Distributive property in geometry Is the equality correct? Matching expressions equal in value Solving an equation using the extended distributive law Solving the problem Using additional geometric shapes Using a rectangle Using variables Applying the formula Complete the missing number Decomposing a Number into smaller numbers/parts Extracting a Common Factor from a group of numbers following decomposition Identifying a Common Factor of a number following decomposition Matching expressions equal in value Number of terms Solving the problem Third power

Algebraic Method Practice Problems with Solutions

Understanding Algebraic Technique

Algebraic Method

Distributive Property

Extended Distributive Property

Factoring

Practice Algebraic Technique

Examples with solutions for Algebraic Technique

Frequently Asked Questions

What is the distributive property in algebra?

How do you factor algebraic expressions?

What's the difference between (-4)² and -4²?

How does the extended distributive property work?

When should I use factoring vs distributive property?

What are common mistakes with exponents in algebra?

How do algebraic methods help solve equations?

What grade level learns these algebraic techniques?

More Algebraic Technique Questions

Extended Distributive Property

Factorization - Common Factor

Practice by Question Type

More Resources and Links