Problem 31 Factor completely. If a polynomi... [FREE SOLUTION]

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Chapter 6: Problem 31

Factor completely. If a polynomial is prime, state this. $$ -8 t^{2}-8 t+30 $$

Short Answer

Expert verified

The factorization is (x - y)(x^2 + xy + y^2).

Step by step solution

Identify the difference of cubes

Recognize that the given polynomial, ewline (x^3 - y^3),is a difference of cubes.

Recall the difference of cubes formula

The general formula for factoring a difference of cubes is: ewline (x^3 - y^3) = (x - y)(x^2 + xy + y^2).

Apply the difference of cubes formula

Substitute (x) and (y) into the formula to get: ewline (x^3 - y^3) = (x - y)(x^2 + xy + y^2).

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Key Concepts

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

Factoring Polynomials

Polynomials are expressions that contain variables, coefficients, and exponents. Factoring involves breaking down a complex polynomial into simpler expressions, or factors, that when multiplied together, give the original polynomial.
Understanding the structure and terms in the polynomial is crucial for successful factoring. In this case, we deal with a specific type - the difference of cubes.
To factor polynomials like something raised to the power of 3, you need specific formulas (like the difference of cubes) and techniques. The difference of cubes formula is one of these handy tools.

Algebraic Formulas

In algebra, certain formulas help us simplify and solve problems more easily. One such important formula is the *difference of cubes* formula.
The difference of cubes formula is: ewline (x^3 - y^3) = (x - y)(x^2 + xy + y^2).
This formula helps us break down the expression into two simpler binomials and a trinomial. Understanding and memorizing these formulas can remarkably streamline your algebraic manipulations.
Using this simplifies the expressions and makes further mathematical operations easier.

Polynomial Factorization

Polynomial factorization, like the example given, involves breaking down a polynomial into simpler factors. The general method is first to identify the type of polynomial.
In our case, we identified the polynomial as a difference of cubes, and then we applied the difference of cubes formula: ewline (x^3 - y^3) = (x - y)(x^2 + xy + y^2).
This leads to a result with simpler polynomials multiplied together.
Substituting the variables into the equation, we get the final expression: (x^3 - y^3) = (x - y)(x^2 + xy + y^2)
Learning these methods allows you to confidently tackle different polynomial factors.

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91影视

Factor completely. If a polynomial is prime, state this. $$ -8 t^{2}-8 t+30 $$

Short Answer

Step by step solution

Identify the difference of cubes

Recall the difference of cubes formula

Apply the difference of cubes formula

Key Concepts

Factoring Polynomials

Algebraic Formulas

Polynomial Factorization

One App. One Place for Learning.

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