Question 1

What is factoring trinomials?

Accepted Answer

Factoring trinomials is the process of expressing a quadratic trinomial (ax² + bx + c) as a product of two binomial factors. For example, x² + 5x + 6 factors as (x + 2)(x + 3). This is the reverse of expanding binomials and is essential for solving quadratic equations and simplifying algebraic expressions.

Question 2

How do you factor trinomials?

Accepted Answer

To factor trinomials: 1) If a=1, find two numbers that multiply to c and add to b. 2) For a≠1, use the AC method: find two numbers that multiply to ac and add to b, then group terms. 3) Factor by grouping. 4) Check by expanding the factors. For perfect square trinomials, use (a±b)² patterns.

Question 3

What is the AC method for factoring trinomials?

Accepted Answer

The AC method factors trinomials ax² + bx + c by: 1) Find two numbers that multiply to a×c and add to b. 2) Split the middle term bx using these numbers. 3) Group the four terms into two pairs. 4) Factor out the GCF from each pair. 5) Factor out the common binomial. This method works for any quadratic trinomial.

Question 4

What are perfect square trinomials?

Accepted Answer

Perfect square trinomials have the form a² ± 2ab + b² = (a ± b)². Examples: x² + 6x + 9 = (x + 3)², 4x² - 12x + 9 = (2x - 3)². They can be recognized when the first and last terms are perfect squares and the middle term is twice the product of the square roots.

Question 5

How do you factor trinomials with a leading coefficient not equal to 1?

Accepted Answer

For ax² + bx + c where a ≠ 1: Use the AC method. Find two numbers p and q where p×q = a×c and p+q = b. Rewrite bx as px + qx, then group (ax² + px) + (qx + c). Factor each group and then factor out the common binomial. For example, 6x² + 17x + 5 factors as (2x + 5)(3x + 1).

Question 6

When can trinomials not be factored?

Accepted Answer

Trinomials cannot be factored over the integers when no pair of integers multiply to ac and add to b. They also cannot be factored over real numbers if the discriminant (b² - 4ac) is negative. In such cases, the trinomial may require the quadratic formula or complex factorization techniques.

Question 7

What is the difference between factoring and solving quadratic equations?

Accepted Answer

Factoring expresses a trinomial as a product: x² + 5x + 6 = (x + 2)(x + 3). Solving uses the factored form to find roots: (x + 2)(x + 3) = 0 when x = -2 or x = -3. Factoring is a tool used in solving, along with completing the square and the quadratic formula.

Question 8

How do you check if factoring is correct?

Accepted Answer

To verify factoring, expand the factors using FOIL or distribution. If (x + 2)(x + 3) is expanded, you should get x² + 5x + 6. Multiply: x² + 3x + 2x + 6 = x² + 5x + 6. If the expanded form matches the original trinomial, the factoring is correct.

Question 9

What is factoring by grouping?

Accepted Answer

Factoring by grouping is used when a polynomial has four terms. Group terms into two pairs, factor out the GCF from each pair, then factor out the common binomial. For example, x³ + 3x² + 2x + 6 = x²(x + 3) + 2(x + 3) = (x + 3)(x² + 2). This method works with the AC method for trinomials.

Question 10

Why is factoring trinomials important?

Accepted Answer

Factoring is crucial for: solving quadratic equations, simplifying rational expressions, finding zeros of functions, graphing parabolas, analyzing quadratic functions, solving optimization problems, and understanding polynomial behavior. It's fundamental to algebra and calculus applications.

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