Problem 49 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ z(y+4)+3(y+4) $$

Step-by-Step Solution

Answer

The GCF of the expression is \((y+4)\), and the factored form is \((y+4)(z+3)\).

Step 1: Identify Common Factor

Observe the expression: \[z(y+4)+3(y+4) \]Notice that \((y+4)\) is a common factor in both terms.

Step 2: Factor Out the GCF

Factor out the common factor \((y+4)\) from the expression:\[(y+4)(z+3)\]When you factor \((y+4)\) out of \(z(y+4)\), you are left with \(z\), and when you factor it out of \(3(y+4)\), you are left with \(3\).

Step 3: Verify the Factoring

Expand the factored expression to confirm:\[(y+4)(z+3) = y(z+3) + 4(z+3) = yz + 3y + 4z + 12\]This should match the original expression \(z(y+4) + 3(y+4)\) when expanded again. Thus, the factoring is correct.

Key Concepts

GCF (Greatest Common Factor)

When working with polynomial expressions, one of the first steps in simplifying or factoring them is to identify the Greatest Common Factor (GCF). The GCF of terms in a polynomial is the largest expression that divides each term without leaving a remainder.

Understanding the GCF is crucial because it helps reveal the structure of the polynomial, making it easier to work with. In the expression \(z(y+4) + 3(y+4)\), the parenthetical \((y+4)\) is present in both terms, making it the GCF.

• To determine the GCF, look for repeating expressions or constants in each term.
• Factor out the GCF, simplifying the polynomial expression further.

Recognizing the GCF is important in performing operations on polynomials, such as addition or multiplication, where common factors simplify the process. By factoring out the GCF, students can solve polynomial equations more efficiently.

Factoring Process

Factoring is a kind of reverse engineering of multiplication involving polynomials. It helps to break down complex expressions into simpler, more manageable parts.

In the factoring process, you aim to rewrite the polynomial as a product of its factors. This involves:

• Identifying a common factor across terms, such as the GCF.
• Dividing each term by this GCF and writing it outside the parentheses.
• Constructing the remaining expression inside the parentheses.

For the polynomial \(z(y+4) + 3(y+4)\), after recognizing \((y+4)\) as the GCF, the polynomial can be rewritten as \((y+4)(z+3)\).

By mastering the factoring process, students can simplify problems and reveal solutions more clearly. Factoring also plays a key role in solving polynomial equations, as it can transform them into a format suitable for setting individual terms equal to zero to find roots.

Polynomial Expressions

A polynomial expression is a mathematical phrase involving a sum of powers in one or more variables multiplied by coefficients. For example, the expression \(z(y+4) + 3(y+4)\) is a polynomial where both terms share a common factor (\(y+4\)).

Understanding polynomial expressions requires recognizing the components, such as:

• Terms: These are the distinct parts of the expression separated by addition or subtraction, like \(z(y+4)\) and \(3(y+4)\).
• Coefficients: These are the numbers that multiply the variables, such as 1 for \(z\) and 3 for the second term.
• Variables: The letters that represent numbers, like \(y\) and \(z\) in this case.

Polynomial expressions form the foundation of much of algebra and are crucial in various applications from simple equations to more complex calculus problems. By understanding their structure, students can manipulate and simplify these expressions, making algebra more approachable and less intimidating.