Problem 32 Factor completely. If a polynomi... [FREE SOLUTION]

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Chapter 6: Problem 32

Factor completely. If a polynomial is prime, state this. $$ -36 a^{2}+21 a-3 $$

Short Answer

Expert verified

The factored form is $(2t + 1)(4t^2 - 2t + 1)$.

Step by step solution

Identify the form of the polynomial

Recognize that the given polynomial, $8t^3 + 1$, is a sum of cubes. A sum of cubes has the form $a^3 + b^3$.

Determine $a$ and $b$

Rewrite the polynomial in the form $a^3 + b^3$ where $a = 2t$ and $b = 1$. Thus, $8t^3 = (2t)^3$ and $1 = 1^3$.

Apply the sum of cubes formula

Use the sum of cubes factoring formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. In this case, substitute $a = 2t$ and $b = 1$ into the formula: i) $(2t + 1)$ ii) $(2t)^2 - (2t)(1) + (1)^2$

Simplify the factors

Simplify the expressions: i) $(2t + 1)$ remains as it is ii) $(2t)^2 = 4t^2$, $-(2t)(1) = -2t$, and $(1)^2 = 1$So, the second factor simplifies to: $4t^2 - 2t + 1$

Write the final factored form

Combine the simplified expressions: $8t^3 + 1 = (2t + 1)(4t^2 - 2t + 1)$.

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Key Concepts

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

Sum of Cubes

When dealing with polynomials, recognizing special forms can make factoring easier. One such form is the 'sum of cubes.' A sum of cubes looks like this: $a^3 + b^3$.
In these polynomials, the terms represent perfect cubes. For example, $8t^3$ is the cube of $2t$ because $(2t)^3 = 8t^3$. Similarly, 1 is the cube of 1 since $1^3 = 1$.
By identifying the sum of cubes $8t^3 + 1$ as $(2t)^3 + 1^3$, we set the stage for easy factoring using formulas.

Factoring Formula

To factor sums of cubes, we use a specific formula:
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
This formula provides a straightforward way to break down the polynomial into simpler parts.
For our given polynomial $8t^3 + 1$, we identified $a = 2t$ and $b = 1$.
Plugging these values into the formula results in:
$(2t + 1)( (2t)^2 - (2t)(1) + (1)^2 )$.
Simplifying the second term gives us $4t^2 - 2t + 1$.
Therefore, the factored form is $ (2t + 1)(4t^2 - 2t + 1) $.

Algebraic Expressions

Understanding algebraic expressions is essential in factoring polynomials. An algebraic expression combines variables, constants, and arithmetic operations.
Our polynomial $8t^3 + 1$ is an algebraic expression consisting of two terms.
When factoring, we often rewrite expressions to identify patterns or forms like the sum of cubes.
Simplifying expressions is key. For instance, reducing $(2t)^2$ to $4t^2$ or $ (2t)(1) $ to $ 2t $ breaks down complex expressions into more manageable parts.
Identifying these parts allows us to clearly write the factored form.

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91影视

Factor completely. If a polynomial is prime, state this. $$ -36 a^{2}+21 a-3 $$

Short Answer

Step by step solution

Identify the form of the polynomial

Determine \(a\) and \(b\)

Apply the sum of cubes formula

Simplify the factors

Write the final factored form

Key Concepts

Sum of Cubes

Factoring Formula

Algebraic Expressions

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