Question
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Answer
26)
28)
30)
32)
28)
30)
32)
Solution
26) Factor
- Notice that each term has a common factor of
. Factor it out: - Recognize that the quadratic
is a perfect square: - Hence, the complete factorization is:
28) Factor
- First, factor out the greatest common factor, which is
: - Now, factor the quadratic
. Look for two numbers that multiply to and add to (these are and ): - Therefore, the complete factorization is:
30) Factor
- Notice that each term has a common factor of
. Factor it out: - Factor the quadratic
. Look for two numbers that multiply to (with the present, consider them as coefficients along with and ) and add to . The suitable pair is and because and . - Rewrite the middle term using these numbers:
- Factor by grouping:
- Group the first two terms and the last two terms:
- Factor out
from the first group and from the second group: - Notice the common factor
:
- Group the first two terms and the last two terms:
- Reintroduce the common factor
:
32) Factor
- Factor the quadratic
. Look for two numbers that multiply to and add up to . The numbers and satisfy this because and . - Rewrite the middle term using these numbers:
- Factor by grouping:
- Group the first two terms and the last two terms:
- Factor out
from the first group and from the second group: - Notice the common factor
:
- Group the first two terms and the last two terms:
- Therefore, the complete factorization is:
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Beyond the Answer
Did you know that polynomial expressions like those you’ve mentioned can often be factored to simplify calculations? For example,
can be factored by taking out the greatest common factor, which is
, leaving you with
. This not only makes the expression easier to handle but reveals more about its structure!
When tackling these polynomials, a common mistake is skipping the step of identifying the greatest common factor (GCF) before attempting to factor completely. Always remember to look for a GCF first, as this can drastically simplify the process. For example, in
, if you factor out a
, you’re left with
, making your life much easier in the long run!