91Ó°ÊÓ

Factoring polynomials often involves identifying the greatest common factor (GCF) in expressions. The GCF is the largest term that divides each term within the polynomial without a remainder.
In our exercise, we're given the polynomial \( x(x+7)+10(x+7) \). Notice that \( x+7 \) is a common factor in both terms \( x(x+7) \) and \( 10(x+7) \).
When factoring, always look for terms or expressions that repeat across your polynomial. Pulling out this common element can greatly simplify the equation. By identifying \( x+7 \) as the GCF, we can factor it out and rewrite the polynomial in a simpler form: \( (x+7)(x+10) \).
Understanding how to find the GCF is crucial in the process of simplifying polynomials and making complex algebraic expressions manageable.

A binomial is a polynomial with exactly two terms, typically joined by a plus or minus sign. For example, \( (x+7) \) in our exercise is a binomial because it consists of two terms: \( x \) and \( 7 \).
Binomials can often be factored out of polynomials when they appear as a common element across multiple terms. This is what we see in the polynomial \( x(x+7) + 10(x+7) \), where \( (x+7) \) is a binomial common to both terms.

• Recognizing binomials in expressions helps in identifying possible factors and performing successful algebraic manipulations.

Learning to spot and factor these can make working with algebraic equations much more straightforward.

The distributive property is a fundamental principle used in algebra to simplify and solve expressions. It states that for any numbers \( a \), \( b \), and \( c \), \( a(b + c) = ab + ac \).
Essentially, you "distribute" \( a \) across each term within the parentheses. In the context of our exercise, it's the basis for the reverse process used to check our work when factoring.
After factoring, expanding \( (x+7)(x+10) \) using the distributive property should give back the original polynomial \( x(x+7)+10(x+7) \).

• First, distribute \( x+7 \) over \( x \) to get \( x(x+7) \), and then over \( 10 \) to get \( 10(x+7) \).

This confirms the factored expression matches the original, highlighting the usefulness of the distributive property in both expansion and factoring of polynomials.