https://file.aiquickdraw.com/mathbot/file/17229683060922dkn1y95.jpg

To solve the problem of matching each polynomial with the most efficient factoring technique, we need to analyze each polynomial and identify the appropriate method for factoring. The techniques available are:

- A: Common factoring
- B: Differences of squares factoring
- C: Trinomial factoring
- D: Factoring by grouping
- E: Factoring using the factor theorem and long or synthetic division

Let's go through each polynomial:

1. \((x - 3)^2 - 49y^2\)

This is a difference of squares: \((a^2 - b^2 = (a - b)(a + b))\).
- Here, \(a = (x - 3)\) and \(b = 7y\).

\[
(x - 3)^2 - 49y^2 = ((x - 3) - 7y)((x - 3) + 7y)
\]

So, the answer is B.

2. \(x^3 + 2x^2 - 5x - 6\)

This polynomial is a cubic polynomial and can be factored using the factor theorem or synthetic division.
- Trying possible rational roots (±1, ±2, ±3, ±6), we find that \(x = 1\) is a root.

Performing synthetic division with \(x - 1\):

\[
\begin{array}{r|rrr}
1 & 1 & 2 & -5 & -6 \\
  &   & 1 &  3 & -2 \\
\hline
  & 1 & 3 & -2 & 0 \\
\end{array}
\]

So, \(x^3 + 2x^2 - 5x - 6 = (x - 1)(x^2 + 3x + 6)\), which can be further factored if possible.
- Since the quadratic does not factor further over the reals, the technique used is E.

3. \(x^2 - 7x - 6\)

This is a trinomial and can be factored using trinomial factoring.
- Finding two numbers that multiply to \(-6\) and add up to \(-7\): \(-1\) and \(-6\).

\[
x^2 - 7x - 6 = (x - 7)(x + 1)
\]

So, the answer is C.

4. \(3x^3 - 11x^2 + 17x\)

We can factor out a common factor, \(x\), first:

\[
3x^3 - 11x^2 + 17x = x(3x^2 - 11x + 17)
\]

The quadratic part cannot be factored further, so the technique used is A.

5. \(3x^2 - 3y^2 - 27\)

We can factor by grouping:

\[
3x^2 - 3y^2 - 27 = 3(x^2 - y^2 - 9) = 3((x - 3)(x + 3) - y^2)
\]

So, the answer is D.

6. \(x^4 - 13x^2 + 36\)

This is a quadratic in terms of \(x^2\):

Let \(u = x^2\), then \(u^2 - 13u + 36 = 0\).

Factoring the quadratic:

\[
u^2 - 13u + 36 = (u - 4)(u - 9)
\]

Returning to \(x\):

\[
(x^2 - 4)(x^2 - 9) = (x - 2)(x + 2)(x - 3)(x + 3)
\]

So, the answer is C.

The final matching is:

1. B (Differences of squares factoring)
2. E (Factoring using the factor theorem and long or synthetic division)
3. C (Trinomial factoring)
4. A (Common factoring)
5. D (Factoring by grouping)
6. C (Trinomial factoring)

Do you have any questions or would you like further details on any of these steps?

Here are 8 related questions to expand on this topic:

1. How do you identify a difference of squares in a polynomial?
2. What are the steps involved in using synthetic division to factor a polynomial?
3. How can the factor theorem be applied to find roots of a polynomial?
4. What is the process of trinomial factoring, and how do you recognize it?
5. When is factoring by grouping an appropriate method?
6. How do you factor a cubic polynomial using the factor theorem?
7. What techniques are used to factor higher degree polynomials?
8. How do you handle polynomials that do not factor over the real numbers?

**Tip:** Always check for a common factor first before applying other factoring techniques.

Learn how to solve polynomial factoring problems with a detailed step-by-step solution. This problem covers key concepts such as synthetic division, trinomial factoring, and difference of squares. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation