Problem 116 Factor out all common factors fi... [FREE SOLUTION]

Chapter 8: Problem 116

Factor out all common factors first including $-1$ if the first term is negative. If an expression is prime, so indicate. $$ -r+3 r^{2}-10 $$

Short Answer

Expert verified

The expression $-r + 3r^2 - 10$ is factored as $-1(3r^2 - r + 10)$, and the quadratic is prime.

Step by step solution

Identify the Leading Coefficient

Look at the expression $ -r + 3r^2 - 10 $. The leading term here is $-r$ because it will take precedence due to the ordering $ r^2 > r^1 > r^0 $. Since this term is negative, we need to factor out $-1$ first to simplify our work.

Factor out -1

Factor $-1$ from the entire expression: $-1(r - 3r^2 + 10)$. Now, we will work with the expression inside the parentheses.

Rearrange the Terms

Rewrite the terms inside the parentheses in order of decreasing powers of $r$: $-1(3r^2 - r + 10)$. This makes it easier to look for any additional factoring possibilities.

Check for a Greatest Common Factor

Examine the new expression $3r^2 - r + 10$ to see if there is a common factor among all terms. In this case, there is no common factor except 1.

Check for Factorability of the Quadratic

Attempt to factor $3r^2 - r + 10$ using methods like splitting the middle term or applying the quadratic formula. This expression does not factor neatly into two binomials, suggesting that it may be prime.

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Key Concepts

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

Leading Coefficient

The leading coefficient is the first coefficient you encounter when writing a polynomial in standard form, which is from highest to lowest degree. For the expression

a leading term like $-r$ indicates that there is a coefficient of $-1$ in front of the variable $r$.
Because it is negative, the step-by-step solution advises to factor out $-1$. This slight adjustment can significantly simplify the problem-solving process.

It prepares the expression for easier manipulation by potentially exposing other factorizable components, making the subsequent steps cleaner and manageable. Remember, whenever you encounter a negative lead, pulling out $-1$ is a great first strategy in any factoring journey.

Common Factor

Identifying a common factor is crucial in breaking down expressions into simpler components. Consider the adjusted expression after the lead is reworked: $3r^2 - r + 10$.

Primarily, you'll want to check each term for shared factors.
For this expression, there are no additional common factors present other than $1$.
Recognizing when the only common factor is $1$ indicates that no further basic simplification is possible with that method.

Recognizing this allows you to efficiently decide on which methodology to apply next (e.g., quadratic techniques for further examination). Therefore, verifying common factors is an essential consideration during factoring, keeping your approach focused and efficient.

Quadratic Expression

Quadratic expressions take the form $ax^2 + bx + c$.These expressions are characterized by a degree of 2, meaning $r^2$terms are included. Each term鈥檚 cooperation is crucial for any factoring process:

If factorable, quadratic expressions can be broken into pairs of binomials.
In $3r^2 - r + 10 $, because $a, b, and c$ don't meet simple factorization conditions, other techniques may be necessary.
These include splitting the middle term or using formulas.

With quadratics, factorization isn't always straightforward, demanding an evaluative step-by-step method to reveal simplified forms or confirm no possible decomposition.

Prime Expression

When an expression resists breaking down into simpler multipliers, it's termed a prime expression. Prime expressions are like indivisible numbers in mathematics, only they can't be further reduced or factored over the integers.

Testing for primality involves checking various factorization methods.
If none of these methods breaks the expression further, like in $3r^2 - r + 10$, which remains unchanged through standard checks.
The expression qualifies as prime.

Acknowledging an expression as prime confirms its integrity as a complete mathematical entity, often marking the end of the factorization quest, ensuring no simpler forms are missed.

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91影视

Factor out all common factors first including \(-1\) if the first term is negative. If an expression is prime, so indicate. $$ -r+3 r^{2}-10 $$

Short Answer

Step by step solution

Identify the Leading Coefficient

Factor out -1

Rearrange the Terms

Check for a Greatest Common Factor

Check for Factorability of the Quadratic

Key Concepts

Leading Coefficient

Common Factor

Quadratic Expression

Prime Expression

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