Problem 43 Factor completely. $$ t^{6}+... [FREE SOLUTION]

Chapter 6: Problem 43

Factor completely. $$ t^{6}+64 y^{6} $$

Short Answer

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(t^2 + 2y^2)(t^4 - 2t^2y^2 + 4y^4)

Step by step solution

Recognize the sum of cubes

First, recognize that the given expression can be rewritten as a sum of cubes. Both terms are perfect cubes: $t^6 = (t^2)^3$ and $64y^6 = (2y^2)^3$. So, the expression becomes: \[(t^2)^3 + (2y^2)^3\]

Apply the sum of cubes formula

Recall the sum of cubes formula: \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]. Here, $a = t^2$ and $b = 2y^2$. Substitute these into the formula to get: \[(t^2 + 2y^2)(t^4 - t^2 \times 2y^2 + (2y^2)^2)\]

Simplify the expression

Next, simplify the terms inside the second parenthesis: \[t^4 - t^2 \times 2y^2 + (2y^2)^2\]. This simplifies to: \[t^4 - 2t^2y^2 + 4y^4\]. So the fully factored form is: \[(t^2 + 2y^2)(t^4 - 2t^2y^2 + 4y^4)\]

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Key Concepts

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

Sum of Cubes

The 'sum of cubes' is a polynomial expression that takes the form: \[a^3 + b^3\].
In these cases, both terms are perfect cubes.
For instance, in the expression from the exercise, we can identify: $t^6$ and $64y^6$ as perfect cubes.
This allows us to rewrite $t^6$ as $(t^2)^3$and $64y^6$ as $(2y^2)^3$.We can apply the sum of cubes formula once the expression is rewritten.
This is a crucial step in factoring such polynomial expressions.

Factoring Techniques

There are various factoring techniques for polynomial expressions.
One important method is factoring by recognizing special forms like the sum of cubes.
We use the sum of cubes formula, which is:$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
In this specific problem, we identify $a = t^2$ and $b = 2y^2$ from the rewritten expression $(t^2)^3 + (2y^2)^3$.
Substituting these into the formula provides the factored form:$(t^2 + 2y^2)(t^4 - 2t^2y^2 + 4y^4)$.
Applying this factoring technique helps simplify polynomial expressions into products of simpler polynomials.

Polynomial Expressions

Polynomial expressions can have varying degrees and forms.
Recognizing common factoring patterns helps in simplifying them.
Polynomials can have terms that need special factoring:

Difference of squares
Sum of cubes
Difference of cubes

In the given exercise, the polynomial $t^6 + 64y^6$ appears complex.
But by recognizing it as a sum of cubes, we can rewrite and factor it using special formulas.
Factoring reduces complicated polynomials into simpler factors.
This is useful in solving polynomial equations and in various algebraic manipulations.

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91影视

Factor completely. $$ t^{6}+64 y^{6} $$

Short Answer

Step by step solution

Recognize the sum of cubes

Apply the sum of cubes formula

Simplify the expression

Key Concepts

Sum of Cubes

Factoring Techniques

Polynomial Expressions

One App. One Place for Learning.

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