Problem 2 - Elementary Algebra

Question

Factor, if possible, the following trinomials. \(y^{2}+7 y-30\)

Step-by-Step Solution

Answer

Answer: The factored form is \((y + 10)(y - 3)\).

Step 1: Write down the given trinomial

Write the given trinomial which is \(y^2 + 7y - 30\).

Step 2: Find two numbers that multiply to -30 and add up to 7

We are looking for two numbers whose product is -30 and whose sum is 7. After trying different combinations, we find that 10 and -3 satisfy these conditions: \(10 \times (-3) = -30\) and \(10 + (-3) = 7\).

Step 3: Rewrite the middle term using the two numbers found in step 2

We found two numbers 10 and -3, and we will use them to rewrite the given trinomial by splitting the middle term into two terms: \(y^2 + 10y - 3y - 30\).

Step 4: Factor by grouping

Now, we will group the terms in pairs and factor out the greatest common factor (GCF) from each pair: \((y^2 + 10y) + (-3y - 30)\). For the first pair, the GCF is \(y\) and for the second pair, the GCF is -3. So, after factoring, we get: \(y(y + 10) - 3(y + 10)\).

Step 5: Factor the common binomial

Notice that now there is a common binomial factor, \((y + 10)\), in both terms. We can factor it out from the whole expression to get: \((y + 10)(y - 3)\). The factored form of the given trinomial is \((y + 10)(y - 3)\).

Key Concepts

Polynomials

Polynomials are fundamental algebraic expressions. They consist of variables raised to various powers, multiplied by coefficients. The standard form of a polynomial arranges these terms in descending order of power. For example, in the polynomial \(y^2 + 7y - 30\), we have:

• The term \(y^2\), where 2 is the highest power of \(y\), also known as the degree of the polynomial.
• The term \(7y\), where the coefficient of \(y\) is 7.
• The constant term -30, which does not have a variable associated with it.

Understanding polynomials is crucial because they form the basis of more complex algebraic expressions. Trinomials like \(y^2 + 7y - 30\) are special polynomials consisting of three terms. Factoring polynomials is an essential skill that simplifies these expressions and solves polynomial equations.

Factoring by Grouping

Factoring by grouping is a useful technique for solving polynomial equations, particularly those with four terms. In the case of trinomials, this technique can be adapted by splitting the middle term, just like in the exercise we are discussing.When presented with the trinomial \(y^2 + 7y - 30\), the process begins by finding two numbers that multiply to the trailing constant term (-30) and add up to the middle coefficient (7). These numbers are 10 and -3.

• Rewrite the trinomial by splitting the middle term to create four terms: \(y^2 + 10y - 3y - 30\).
• Group the terms into two pairs: \((y^2 + 10y) + (-3y - 30)\).
• Factor the greatest common factor from each pair: factor \(y\) from \(y^2 + 10y\) and -3 from \(-3y - 30\).

This method relies on identifying and factoring common binomials, simplifying the expression, and ultimately leading to the factored form of the polynomial.By practicing this technique, students can approach other polynomial expressions with confidence and creativity.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent mathematical quantities. They can range from simple expressions, such as \(3x + 2\), to complex polynomials with multiple variables and terms, like \(x^2 - 4xy + y^2 + 11\). These expressions can often be manipulated through operations such as addition, subtraction, multiplication, division, and factoring.In the specific case of trinomials like \(y^2 + 7y - 30\), they require particular techniques for simplification and solving, like factoring.

• Factoring is used to express the expression as a product of simpler expressions, making them more manageable for solving equations or further simplification.
• Mastering these techniques translates to a better understanding of algebra as a whole.
• Understanding the components and structure of algebraic expressions is crucial for effectively applying operations like grouping and factoring.

Consistently practicing these techniques will enhance one's ability to solve real-world problems involving algebraic expressions, making algebra a highly applicable and valuable field.