Factor out the GCF from each polynomial. See Examples 4 through 10. (2)/(5) y^7-(4)/(5) y^5+(3)/

Factor out the GCF from each polynomial. See Examples 4...

Key Concepts

Fraction Simplification

Fraction simplification involves reducing fractions by removing common numerical factors from the numerator and the denominator. When factoring polynomials with fractional coefficients, it is important to treat the rational part correctly by identifying and extracting the common numerical fraction from each term.

Properties of Exponents

The properties of exponents govern how powers of variables multiply and divide. When factoring out a common variable factor from polynomial terms, the rules of exponents ensure that the correct power is extracted, and the remaining terms have their exponents adjusted accordingly. This is crucial for maintaining the integrity of the original expression once the GCF has been factored out.

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides each term of a polynomial without a remainder. It combines both numerical and variable factors common to all terms, and factoring it out simplifies the expression and makes further manipulation or factorization easier.

Polynomial Factoring

Polynomial factoring is the process of rewriting a polynomial as a product of simpler polynomials or monomials. It is a fundamental algebraic skill that helps in solving equations, simplifying expressions, and analyzing functions. In this context, extracting the GCF is one of the first steps in polynomial factoring.

Transcript

00:01 So for this problem, we are trying to find the gcf of our polynomial equation.

00:05 First thing i want to do is go ahead and highlight all of my constants versus all of my variables.

00:12 And if there is a negative sign out front, i'm highlighting that too.

00:20 So i've gone ahead and done that.

00:22 Now i can easily see what needs to pair up with what.

00:25 When i look at my fractions, i can easily find them.

00:28 Same thing with my variables.

00:30 When i'm looking at my variable, remember we need to take the least, the smallest number that goes into all three, we could go ahead and work this out.

00:38 However, looking at this easily, i see the only logical answer will be one.

00:44 One.

00:45 We'll go ahead and write that off to the side and finish this.

00:49 Two -fifths, y to the sixth, because remember we add our exponents when we multiply variables together.

00:57 Four -thirds, y to the fourth, four -fits, i'm sorry, three -fifths, y to the first power, which i can just leave alone, in two -fifths.

01:10 Now we need to go ahead and work with our fractions.

01:16 Since our denominators are all the same, we need to look at our numerators.

01:20 So we have a two, a four, a three, and a two...

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