Problem 65 Factor If $Q(x)=x^{6},$ use ... [FREE SOLUTION]

91影视

Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 5: Problem 65

Factor If $Q(x)=x^{6},$ use factoring to simplify $$ Q(a+h)-Q(a) $$

Short Answer

Expert verified

h(6a^5 + 15a^4h + 20a^3h^2 + 15a^2h^3 + 6ah^4 + h^5)

Step by step solution

Write the expression

Start with the given expression: \[Q(a+h)-Q(a)\]

Substitute the given function

Since $Q(x) = x^6$, replace $Q(a+h)$ and $Q(a)$ with their corresponding values: \[Q(a+h) = (a+h)^6\] \[Q(a) = a^6\]

Simplify the expression

Substitute $Q(a+h)$ and $Q(a)$ into the expression: \[(a+h)^6 - a^6\]

Apply the Binomial Theorem

Use the binomial theorem to expand $(a+h)^6$: \[(a+h)^6 = a^6 + 6a^5h + 15a^4h^2 + 20a^3h^3 + 15a^2h^4 + 6ah^5 + h^6 \]Substitute this into the expression: \[ (a^6 + 6a^5h + 15a^4h^2 + 20a^3h^3 + 15a^2h^4 + 6ah^5 + h^6) - a^6 \]

Simplify further

Combine like terms to simplify: \[6a^5h + 15a^4h^2 + 20a^3h^3 + 15a^2h^4 + 6ah^5 + h^6\]

Factor out the common term

Notice that every term has at least one $h$ in them. Factor $h$ out: \[ h(6a^5 + 15a^4h + 20a^3h^2 + 15a^2h^3 + 6ah^4 + h^5) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Theorem

The binomial theorem helps us expand expressions that are raised to a power. It states that for any integers, $n $ and any real numbers, $x $ and $y $, the expansion of $(x+y)^n$ is given by:
\[ (x + y)^n = \binom{n}{0} x^n y^0 + \binom{n}{1} x^{n-1} y^1 + \binom{n}{2} x^{n-2} y^2 + \txt{...} + \binom{n}{n} x^0 y^n \]
Here, $ \binom{n}{k} $ stands for the binomial coefficient and can be calculated as:
\[ \binom{n}{k} = \frac {n!}{k! (n-k)!} \]
This theorem is particularly useful when dealing with polynomial expansions, as in the given exercise. Applying the binomial theorem, we expanded $ (a+h)^6 $ to get a detailed list of terms.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial raised to a power as a sum of simpler terms. By expanding $ (a+h)^6 $, we derived:
\[ a^6 + 6a^5h + 15a^4h^2 + 20a^3h^3 + 15a^2h^4 + 6ah^5 + h^6 \]
Every term in this expansion is a product of powers of $a $ and $h $. This kind of expansion is fundamental in both algebra and calculus, where breaking down complex expressions helps simplify and analyze them. Once expanded, we substituted and simplified further by combining like terms.

Common Factor

Finding the common factor in a polynomial helps simplify expressions significantly. In the exercise, after expanding and simplifying the expression $ (a+h)^6 - a^6$, we noticed a common term $h$ in each term:
\[ 6a^5h + 15a^4h^2 + 20a^3h^3 + 15a^2h^4 + 6ah^5 + h^6 \]
Factoring out the common term, we get:
\[ h(6a^5 + 15a^4h + 20a^3h^2 + 15a^2h^3 + 6ah^4 + h^5) \]
This makes the expression much simpler and easier to work with. Identifying and factoring out a common factor is a key skill in algebra that can make solving problems more manageable.

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91影视

Factor If \(Q(x)=x^{6},\) use factoring to simplify $$ Q(a+h)-Q(a) $$

Short Answer

Step by step solution

Write the expression

Substitute the given function

Simplify the expression

Apply the Binomial Theorem

Simplify further

Factor out the common term

Key Concepts

Binomial Theorem

Polynomial Expansion

Common Factor

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