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 $x - 1$:

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.