91Ó°ÊÓ

When dealing with polynomials, identifying common factors is an essential first step. A common factor is something that can be divided out of each term in the polynomial. Look for numbers, variables, or expressions that are present in each term. In our example, the polynomial is given as \(4x(x+8) - 5(x+8)\). Here, we notice that both terms, \(4x(x+8)\) and \(-5(x+8)\), share the common factor \((x+8)\).
By identifying the common factor, we can simplify the expression. This process reduces the complexity of the polynomial. It's crucial to check both coefficients and variable expressions for anything they might have in common. Once identified, you can "take out" this common factor, effectively simplifying the polynomial and setting up the next steps for solving or simplifying further.

Simplifying a polynomial involves reducing it to its simplest form by factoring out any common elements. After identifying the common factor \((x+8)\) in our expression \(4x(x+8) - 5(x+8)\), we can factor it out completely. This means we rewrite the polynomial as a product involving this common factor.

• Begin by removing the common factor \((x+8)\) from each term.
• From the term \(4x(x+8)\), removing \((x+8)\) leaves \(4x\).
• From the term \(-5(x+8)\), it leaves \(-5\).

The simplified polynomial now becomes \((x+8)(4x-5)\).
Simplification not only makes equations easier to work with but also provides a clearer path to solving or factoring more fully if needed. This step often clarifies the overall structure of the polynomial.

Once you have factored a polynomial, verification is crucial. It confirms that the factorization is correct and you didn't make an error along the way. To verify, you expand the factors back into their original form to see if they match the original polynomial.
In our case, we simplified \(4x(x+8) - 5(x+8)\) to \((x+8)(4x-5)\). To check this, perform the multiplication:

• Distribute \((x+8)\) across \((4x-5)\).
• \((x+8) \times 4x = 4x(x+8)\) leads you back to the first term.
• \((x+8) \times -5 = -5(x+8)\) takes you back to the second term.

When these results match the original polynomial,\(4x(x+8) - 5(x+8)\), your factorization is confirmed. Verification ensures accuracy and boosts confidence, especially in tests and exams where correct answers matter.